Theorem fvixp 8960

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 8959	. . 3 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2	1	simp3bi 1147	. 2 ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
3		fveq2 6920	. . . 4 ⊢ (𝑥 = 𝐶 → (𝐹‘𝑥) = (𝐹‘𝐶))
4		fvixp.1	. . . 4 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷)
5	3, 4	eleq12d 2838	. . 3 ⊢ (𝑥 = 𝐶 → ((𝐹‘𝑥) ∈ 𝐵 ↔ (𝐹‘𝐶) ∈ 𝐷))
6	5	rspccva 3634	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷)
7	2, 6	sylan 579	1 ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   Fn wfn 6568  ‘cfv 6573  Xcixp 8955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-ixp 8956

This theorem is referenced by:  funcf2  17932  funcpropd  17967  natcl  18021  natpropd  18046  finixpnum  37565  hspdifhsp  46537