Theorem fvco2

Description: Value of a function composition. Similar to second part of Theorem 3H of Enderton p. 47. (Contributed by NM, 9-Oct-2004) (Proof shortened by Andrew Salmon, 22-Oct-2011) (Revised by Stefan O'Rear, 16-Oct-2014)

	Ref	Expression
Assertion	fvco2	⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		imaco	⊢ ( ( 𝐹 ∘ 𝐺 ) “ { 𝑋 } ) = ( 𝐹 “ ( 𝐺 “ { 𝑋 } ) )
2		fnsnfv	⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → { ( 𝐺 ‘ 𝑋 ) } = ( 𝐺 “ { 𝑋 } ) )
3	2	imaeq2d	⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 “ { ( 𝐺 ‘ 𝑋 ) } ) = ( 𝐹 “ ( 𝐺 “ { 𝑋 } ) ) )
4	1 3	eqtr4id	⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) “ { 𝑋 } ) = ( 𝐹 “ { ( 𝐺 ‘ 𝑋 ) } ) )
5	4	eleq2d	⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( 𝑥 ∈ ( ( 𝐹 ∘ 𝐺 ) “ { 𝑋 } ) ↔ 𝑥 ∈ ( 𝐹 “ { ( 𝐺 ‘ 𝑋 ) } ) ) )
6	5	iotabidv	⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( ℩ 𝑥 𝑥 ∈ ( ( 𝐹 ∘ 𝐺 ) “ { 𝑋 } ) ) = ( ℩ 𝑥 𝑥 ∈ ( 𝐹 “ { ( 𝐺 ‘ 𝑋 ) } ) ) )
7		dffv3	⊢ ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( ℩ 𝑥 𝑥 ∈ ( ( 𝐹 ∘ 𝐺 ) “ { 𝑋 } ) )
8		dffv3	⊢ ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( ℩ 𝑥 𝑥 ∈ ( 𝐹 “ { ( 𝐺 ‘ 𝑋 ) } ) )
9	6 7 8	3eqtr4g	⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑋 ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑋 ) ) )