diff --git a/packages/babel-plugin-transform-es2015-function-name/src/index.js b/packages/babel-plugin-transform-es2015-function-name/src/index.js
index ab403e6ec5f1..d0d870377f27 100644
--- a/packages/babel-plugin-transform-es2015-function-name/src/index.js
+++ b/packages/babel-plugin-transform-es2015-function-name/src/index.js
@@ -3,7 +3,7 @@ import nameFunction from "babel-helper-function-name";
 export default function () {
   return {
     visitor: {
-      "ArrowFunctionExpression|FunctionExpression": {
+      FunctionExpression: {
         exit(path) {
           if (path.key !== "value" && !path.parentPath.isObjectProperty()) {
             let replacement = nameFunction(path);
diff --git a/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform-spec/actual.js b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform-spec/actual.js
new file mode 100644
index 000000000000..f56af5c6019b
--- /dev/null
+++ b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform-spec/actual.js
@@ -0,0 +1,5 @@
+// I don't know if this is a bug with arrow-functions spec: true
+// or with function-name, but the functions are missing their names.
+const x = () => x;
+const y = x => x();
+const z = { z: () => y(x) }.z;
diff --git a/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform-spec/expected.js b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform-spec/expected.js
new file mode 100644
index 000000000000..b842455daeea
--- /dev/null
+++ b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform-spec/expected.js
@@ -0,0 +1,16 @@
+var _this = this;
+
+// I don't know if this is a bug with arrow-functions spec: true
+// or with function-name, but the functions are missing their names.
+const x = function () {
+  babelHelpers.newArrowCheck(this, _this);
+  return x;
+}.bind(this);
+const y = function (x) {
+  babelHelpers.newArrowCheck(this, _this);
+  return x();
+}.bind(this);
+const z = { z: function z() {
+    babelHelpers.newArrowCheck(this, _this);
+    return y(x);
+  }.bind(this) }.z;
\ No newline at end of file
diff --git a/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform-spec/options.json b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform-spec/options.json
new file mode 100644
index 000000000000..d66c99f2625e
--- /dev/null
+++ b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform-spec/options.json
@@ -0,0 +1,3 @@
+{
+  "plugins": ["external-helpers", "transform-es2015-function-name", [ "transform-es2015-arrow-functions", { "spec": true } ]]
+}
diff --git a/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform/actual.js b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform/actual.js
new file mode 100644
index 000000000000..c1e980a79f76
--- /dev/null
+++ b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform/actual.js
@@ -0,0 +1,3 @@
+const x = () => x;
+const y = x => x();
+const z = { z: () => y(x) }.z;
diff --git a/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform/expected.js b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform/expected.js
new file mode 100644
index 000000000000..096e218d2d5d
--- /dev/null
+++ b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform/expected.js
@@ -0,0 +1,9 @@
+const x = function x() {
+  return x;
+};
+const y = function y(x) {
+  return x();
+};
+const z = { z: function z() {
+    return y(x);
+  } }.z;
\ No newline at end of file
diff --git a/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform/options.json b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform/options.json
new file mode 100644
index 000000000000..32ac45253fdc
--- /dev/null
+++ b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/function-name/with-arrow-functions-transform/options.json
@@ -0,0 +1,3 @@
+{
+  "plugins": ["external-helpers", "transform-es2015-function-name", "transform-es2015-arrow-functions"]
+}
diff --git a/packages/babel-plugin-transform-es2015-function-name/test/fixtures/issues/5004/actual.js b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/issues/5004/actual.js
new file mode 100644
index 000000000000..c8b74de50e1d
--- /dev/null
+++ b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/issues/5004/actual.js
@@ -0,0 +1,2 @@
+export const x = ({x}) => x;
+export const y = function () {};
diff --git a/packages/babel-plugin-transform-es2015-function-name/test/fixtures/issues/5004/expected.js b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/issues/5004/expected.js
new file mode 100644
index 000000000000..e7a7ae72b464
--- /dev/null
+++ b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/issues/5004/expected.js
@@ -0,0 +1,2 @@
+export const x = ({ x }) => x;
+export const y = function y() {};
\ No newline at end of file
diff --git a/packages/babel-plugin-transform-es2015-function-name/test/fixtures/issues/5004/options.json b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/issues/5004/options.json
new file mode 100644
index 000000000000..f2178b7ef6bd
--- /dev/null
+++ b/packages/babel-plugin-transform-es2015-function-name/test/fixtures/issues/5004/options.json
@@ -0,0 +1,3 @@
+{
+  "plugins": ["external-helpers", "transform-es2015-function-name"]
+}