Theorem cbvixp 8134

Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.)

Hypotheses

Ref	Expression
cbvixp.1	⊢ Ⅎ𝑦𝐵
cbvixp.2	⊢ Ⅎ𝑥𝐶
cbvixp.3	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

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

Distinct variable group:   𝑥,𝐴,𝑦

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

Proof of Theorem cbvixp

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvixp.1	. . . . . 6 ⊢ Ⅎ𝑦𝐵
2	1	nfel2 2924	. . . . 5 ⊢ Ⅎ𝑦(𝑓‘𝑥) ∈ 𝐵
3		cbvixp.2	. . . . . 6 ⊢ Ⅎ𝑥𝐶
4	3	nfel2 2924	. . . . 5 ⊢ Ⅎ𝑥(𝑓‘𝑦) ∈ 𝐶
5		fveq2 6379	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑓‘𝑥) = (𝑓‘𝑦))
6		cbvixp.3	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
7	5, 6	eleq12d 2838	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑓‘𝑥) ∈ 𝐵 ↔ (𝑓‘𝑦) ∈ 𝐶))
8	2, 4, 7	cbvral 3315	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)
9	8	anbi2i 616	. . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶))
10	9	abbii 2882	. 2 ⊢ {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)}
11		dfixp 8119	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
12		dfixp 8119	. 2 ⊢ X𝑦 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐶)}
13	10, 11, 12	3eqtr4i 2797	1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1652   ∈ wcel 2155  {cab 2751  Ⅎwnfc 2894  ∀wral 3055   Fn wfn 6065  ‘cfv 6070  Xcixp 8117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-iota 6033  df-fn 6073  df-fv 6078  df-ixp 8118

This theorem is referenced by:  cbvixpv  8135  mptelixpg  8154  ixpiunwdom  8707  prdsbas3  16423  elptr2  21673  ptunimpt  21694  ptcldmpt  21713  finixpnum  33841  ptrest  33855  hoimbl2  41545  vonhoire  41552  vonn0ioo2  41570  vonn0icc2  41572