Theorem fnfvof

Description: Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014) (Proof shortened by Mario Carneiro, 5-Jun-2015)

	Ref	Expression
Assertion	fnfvof	⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ) ) → ( ( 𝐹 ∘f 𝑅 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑋 ) 𝑅 ( 𝐺 ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		simpll	⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ 𝐴 ∈ 𝑉 ) → 𝐹 Fn 𝐴 )
2		simplr	⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ 𝐴 ∈ 𝑉 ) → 𝐺 Fn 𝐴 )
3		simpr	⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ 𝐴 ∈ 𝑉 ) → 𝐴 ∈ 𝑉 )
4		inidm	⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴
5		eqidd	⊢ ( ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ 𝐴 ∈ 𝑉 ) ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑋 ) )
6		eqidd	⊢ ( ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ 𝐴 ∈ 𝑉 ) ∧ 𝑋 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑋 ) = ( 𝐺 ‘ 𝑋 ) )
7	1 2 3 3 4 5 6	ofval	⊢ ( ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ 𝐴 ∈ 𝑉 ) ∧ 𝑋 ∈ 𝐴 ) → ( ( 𝐹 ∘f 𝑅 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑋 ) 𝑅 ( 𝐺 ‘ 𝑋 ) ) )
8	7	anasss	⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) ∧ ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴 ) ) → ( ( 𝐹 ∘f 𝑅 𝐺 ) ‘ 𝑋 ) = ( ( 𝐹 ‘ 𝑋 ) 𝑅 ( 𝐺 ‘ 𝑋 ) ) )