Theorem cbvixp 8913

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 2943	. . . . 5 ⊢ Ⅎ𝑦(𝑓‘𝑥) ∈ 𝐵
3		cbvixp.2	. . . . . 6 ⊢ Ⅎ𝑥𝐶
4	3	nfel2 2943	. . . . 5 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ 𝐶
5		fveq2 6883	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦))
6		cbvixp.3	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
7	5, 6	eleq12d 2857	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ 𝐵 ↔ (𝑓‘𝑦) ∈ 𝐶))
8	2, 4, 7	cbvralw 3307	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)
9	8	anbi2i 634	. . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶))
10	9	abbii 2830	. 2 ⊢ {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)}
11		dfixp 8898	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
12		dfixp 8898	. 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)}
13	10, 11, 12	3eqtr4i 2796	1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  Ⅎwnfc 2910  ∀wral 3079   Fn wfn 6533  ‘cfv 6538  Xcixp 8896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fn 6541  df-fv 6546  df-ixp 8897

This theorem is referenced by:  mptelixpg  8934  ixpiunwdom  9553  prdsbas3  17535  elptr2  23712  ptunimpt  23733  ptcldmpt  23752  finixpnum  38237  ptrest  38251  hoimbl2  47362  vonhoire  47369  vonn0ioo2  47387  vonn0icc2  47389