Theorem fvixp 8843

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

Hypothesis

Ref	Expression
fvixp.1	⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)

Assertion

Ref	Expression
fvixp	⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝐹

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem fvixp

Step	Hyp	Ref	Expression
1		elixp2 8842	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2	1	simp3bi 1148	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
3		fveq2 6834	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐹‘𝑥) = (𝐹‘𝐶))
4		fvixp.1	. . . 4 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
5	3, 4	eleq12d 2831	. . 3 ⊢ (𝑥 = 𝐶 → ((𝐹‘𝑥) ∈ 𝐵 ↔ (𝐹‘𝐶) ∈ 𝐷))
6	5	rspccva 3564	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷)
7	2, 6	sylan 581	1 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   Fn wfn 6487  ‘cfv 6492  Xcixp 8838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-ixp 8839

This theorem is referenced by:  funcf2  17826  funcpropd  17860  natcl  17914  natpropd  17937  finixpnum  37940  hspdifhsp  47062