Theorem cbvixpv 8863

Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypothesis

Ref	Expression
cbvixpv.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

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

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐶

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

Proof of Theorem cbvixpv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6840	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧‘𝑥) = (𝑧‘𝑦))
2		cbvixpv.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
3	1, 2	eleq12d 2830	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑧‘𝑥) ∈ 𝐵 ↔ (𝑧‘𝑦) ∈ 𝐶))
4	3	cbvralvw 3215	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)
5	4	anbi2i 624	. . 3 ⊢ ((𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) ↔ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶))
6	5	abbii 2803	. 2 ⊢ {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)}
7		dfixp 8847	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
8		dfixp 8847	. 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)}
9	6, 7, 8	3eqtr4i 2769	1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2714  ∀wral 3051   Fn wfn 6493  ‘cfv 6498  Xcixp 8845

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 2708

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fn 6501  df-fv 6506  df-ixp 8846

This theorem is referenced by:  funcpropd  17869  invfuc  17944  natpropd  17946  dprdw  19987  dprdwd  19988  ptuni2  23541  ptbasin  23542  ptbasfi  23546  ptpjopn  23577  ptclsg  23580  dfac14  23583  ptcnp  23587  ptcmplem2  24018  ptcmpg  24022  prdsxmslem2  24494  upixp  38050  rrxsnicc  46728  ioorrnopn  46733  ioorrnopnxr  46735  ovnsubadd  47000  hoidmvlelem4  47026  hoidmvle  47028  hspdifhsp  47044  hoiqssbllem2  47051  hspmbl  47057  hoimbl  47059  opnvonmbl  47062  ovnovollem3  47086