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Theorem cbvixpvw2 36188
Description: Change bound variable and domain in an indexed Cartesian product, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbvixpvw2.1 (𝑥 = 𝑦𝐶 = 𝐷)
cbvixpvw2.2 (𝑥 = 𝑦𝐴 = 𝐵)
Assertion
Ref Expression
cbvixpvw2 X𝑥𝐴 𝐶 = X𝑦𝐵 𝐷
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbvixpvw2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . 7 (𝑥 = 𝑦𝑥 = 𝑦)
2 cbvixpvw2.2 . . . . . . 7 (𝑥 = 𝑦𝐴 = 𝐵)
31, 2eleq12d 2831 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐵))
43cbvabv 2808 . . . . 5 {𝑥𝑥𝐴} = {𝑦𝑦𝐵}
54fneq2i 6662 . . . 4 (𝑡 Fn {𝑥𝑥𝐴} ↔ 𝑡 Fn {𝑦𝑦𝐵})
6 fveq2 6901 . . . . . 6 (𝑥 = 𝑦 → (𝑡𝑥) = (𝑡𝑦))
7 cbvixpvw2.1 . . . . . 6 (𝑥 = 𝑦𝐶 = 𝐷)
86, 7eleq12d 2831 . . . . 5 (𝑥 = 𝑦 → ((𝑡𝑥) ∈ 𝐶 ↔ (𝑡𝑦) ∈ 𝐷))
92, 8cbvralvw2 36169 . . . 4 (∀𝑥𝐴 (𝑡𝑥) ∈ 𝐶 ↔ ∀𝑦𝐵 (𝑡𝑦) ∈ 𝐷)
105, 9anbi12i 627 . . 3 ((𝑡 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑡𝑥) ∈ 𝐶) ↔ (𝑡 Fn {𝑦𝑦𝐵} ∧ ∀𝑦𝐵 (𝑡𝑦) ∈ 𝐷))
1110abbii 2805 . 2 {𝑡 ∣ (𝑡 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑡𝑥) ∈ 𝐶)} = {𝑡 ∣ (𝑡 Fn {𝑦𝑦𝐵} ∧ ∀𝑦𝐵 (𝑡𝑦) ∈ 𝐷)}
12 df-ixp 8931 . 2 X𝑥𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑡𝑥) ∈ 𝐶)}
13 df-ixp 8931 . 2 X𝑦𝐵 𝐷 = {𝑡 ∣ (𝑡 Fn {𝑦𝑦𝐵} ∧ ∀𝑦𝐵 (𝑡𝑦) ∈ 𝐷)}
1411, 12, 133eqtr4i 2771 1 X𝑥𝐴 𝐶 = X𝑦𝐵 𝐷
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1535  wcel 2104  {cab 2710  wral 3057   Fn wfn 6553  cfv 6558  Xcixp 8930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-iota 6510  df-fn 6561  df-fv 6566  df-ixp 8931
This theorem is referenced by: (None)
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