Theorem fvixp

Description: Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016)

	Ref	Expression
Hypothesis	fvixp.1	⊢ ( 𝑥 = 𝐶 → 𝐵 = 𝐷 )
Assertion	fvixp	⊢ ( ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐶 ) ∈ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		fvixp.1	⊢ ( 𝑥 = 𝐶 → 𝐵 = 𝐷 )
2		elixp2	⊢ ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 ↔ ( 𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) )
3	2	simp3bi	⊢ ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )
4		fveq2	⊢ ( 𝑥 = 𝐶 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝐶 ) )
5	4 1	eleq12d	⊢ ( 𝑥 = 𝐶 → ( ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ↔ ( 𝐹 ‘ 𝐶 ) ∈ 𝐷 ) )
6	5	rspccva	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐶 ) ∈ 𝐷 )
7	3 6	sylan	⊢ ( ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐹 ‘ 𝐶 ) ∈ 𝐷 )