Theorem cbvixpv 8974

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 6911	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧‘𝑥) = (𝑧‘𝑦))
2		cbvixpv.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
3	1, 2	eleq12d 2858	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑧‘𝑥) ∈ 𝐵 ↔ (𝑧‘𝑦) ∈ 𝐶))
4	3	cbvralvw 3244	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)
5	4	anbi2i 635	. . 3 ⊢ ((𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) ↔ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶))
6	5	abbii 2830	. 2 ⊢ {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)}
7		dfixp 8958	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)}
8		dfixp 8958	. 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ (𝑧 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑧‘𝑦) ∈ 𝐶)}
9	6, 7, 8	3eqtr4i 2796	1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080   Fn wfn 6556  ‘cfv 6561  Xcixp 8956

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6515  df-fn 6564  df-fv 6569  df-ixp 8957

This theorem is used by:  funcpropd  18109  invfuc  18184  natpropd  18186  dprdw  20259  dprdwd  20260  ptuni2  23928  ptbasin  23929  ptbasfi  23933  ptpjopn  23964  ptclsg  23967  dfac14  23970  ptcnp  23974  ptcmplem2  24405  ptcmpg  24409  prdsxmslem2  24881  upixp  38868  rrxsnicc  47525  ioorrnopn  47530  ioorrnopnxr  47532  ovnsubadd  47797  hoidmvlelem4  47823  hoidmvle  47825  hspdifhsp  47841  hoiqssbllem2  47848  hspmbl  47854  hoimbl  47856  opnvonmbl  47859  ovnovollem3  47883