Theorem cbvixpv 8901

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 6871	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧‘𝑥) = (𝑧‘𝑦))
2		cbvixpv.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
3	1, 2	eleq12d 2859	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑧‘𝑥) ∈ 𝐵 ↔ (𝑧‘𝑦) ∈ 𝐶))
4	3	cbvralvw 3243	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)
5	4	anbi2i 634	. . 3 ⊢ ((𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) ↔ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶))
6	5	abbii 2832	. 2 ⊢ {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)}
7		dfixp 8885	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
8		dfixp 8885	. 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)}
9	6, 7, 8	3eqtr4i 2798	1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {cab 2743  ∀wral 3079   Fn wfn 6520  ‘cfv 6525  Xcixp 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fn 6528  df-fv 6533  df-ixp 8884

This theorem is referenced by:  funcpropd  17949  invfuc  18024  natpropd  18026  dprdw  20073  dprdwd  20074  ptuni2  23694  ptbasin  23695  ptbasfi  23699  ptpjopn  23730  ptclsg  23733  dfac14  23736  ptcnp  23740  ptcmplem2  24171  ptcmpg  24175  prdsxmslem2  24647  upixp  38240  rrxsnicc  46872  ioorrnopn  46877  ioorrnopnxr  46879  ovnsubadd  47144  hoidmvlelem4  47170  hoidmvle  47172  hspdifhsp  47188  hoiqssbllem2  47195  hspmbl  47201  hoimbl  47203  opnvonmbl  47206  ovnovollem3  47230