Theorem cbvixpv 8891

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 6861	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧‘𝑥) = (𝑧‘𝑦))
2		cbvixpv.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
3	1, 2	eleq12d 2823	. . . . 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 623	. . 3 ⊢ ((𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) ↔ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶))
6	5	abbii 2797	. 2 ⊢ {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)}
7		dfixp 8875	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
8		dfixp 8875	. 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)}
9	6, 7, 8	3eqtr4i 2763	1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045   Fn wfn 6509  ‘cfv 6514  Xcixp 8873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fn 6517  df-fv 6522  df-ixp 8874

This theorem is referenced by:  funcpropd  17871  invfuc  17946  natpropd  17948  dprdw  19949  dprdwd  19950  ptuni2  23470  ptbasin  23471  ptbasfi  23475  ptpjopn  23506  ptclsg  23509  dfac14  23512  ptcnp  23516  ptcmplem2  23947  ptcmpg  23951  prdsxmslem2  24424  upixp  37730  rrxsnicc  46305  ioorrnopn  46310  ioorrnopnxr  46312  ovnsubadd  46577  hoidmvlelem4  46603  hoidmvle  46605  hspdifhsp  46621  hoiqssbllem2  46628  hspmbl  46634  hoimbl  46636  opnvonmbl  46639  ovnovollem3  46663