Theorem cbvixp 8595

Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)

Hypotheses

Ref	Expression
cbvixp.1	⊢ Ⅎ𝑦𝐵
cbvixp.2	⊢ Ⅎ𝑥𝐶
cbvixp.3	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvixp	⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbvixp

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvixp.1	. . . . . 6 ⊢ Ⅎ𝑦𝐵
2	1	nfel2 2922	. . . . 5 ⊢ Ⅎ𝑦(𝑓‘𝑥) ∈ 𝐵
3		cbvixp.2	. . . . . 6 ⊢ Ⅎ𝑥𝐶
4	3	nfel2 2922	. . . . 5 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ 𝐶
5		fveq2 6717	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦))
6		cbvixp.3	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
7	5, 6	eleq12d 2832	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ 𝐵 ↔ (𝑓‘𝑦) ∈ 𝐶))
8	2, 4, 7	cbvralw 3349	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)
9	8	anbi2i 626	. . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶))
10	9	abbii 2808	. 2 ⊢ {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)}
11		dfixp 8580	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
12		dfixp 8580	. 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)}
13	10, 11, 12	3eqtr4i 2775	1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  {cab 2714  Ⅎwnfc 2884  ∀wral 3061   Fn wfn 6375  ‘cfv 6380  Xcixp 8578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fn 6383  df-fv 6388  df-ixp 8579

This theorem is referenced by:  cbvixpv  8596  mptelixpg  8616  ixpiunwdom  9206  prdsbas3  16986  elptr2  22471  ptunimpt  22492  ptcldmpt  22511  finixpnum  35499  ptrest  35513  hoimbl2  43878  vonhoire  43885  vonn0ioo2  43903  vonn0icc2  43905