Theorem cbvixpv 8865

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 6842	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧‘𝑥) = (𝑧‘𝑦))
2		cbvixpv.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
3	1, 2	eleq12d 2831	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑧‘𝑥) ∈ 𝐵 ↔ (𝑧‘𝑦) ∈ 𝐶))
4	3	cbvralvw 3216	. . . 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 2804	. 2 ⊢ {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)}
7		dfixp 8849	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
8		dfixp 8849	. 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)}
9	6, 7, 8	3eqtr4i 2770	1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052   Fn wfn 6495  ‘cfv 6500  Xcixp 8847

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fn 6503  df-fv 6508  df-ixp 8848

This theorem is referenced by:  funcpropd  17838  invfuc  17913  natpropd  17915  dprdw  19953  dprdwd  19954  ptuni2  23532  ptbasin  23533  ptbasfi  23537  ptpjopn  23568  ptclsg  23571  dfac14  23574  ptcnp  23578  ptcmplem2  24009  ptcmpg  24013  prdsxmslem2  24485  upixp  37977  rrxsnicc  46655  ioorrnopn  46660  ioorrnopnxr  46662  ovnsubadd  46927  hoidmvlelem4  46953  hoidmvle  46955  hspdifhsp  46971  hoiqssbllem2  46978  hspmbl  46984  hoimbl  46986  opnvonmbl  46989  ovnovollem3  47013