Theorem cbvixpvw2 36224

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 = 𝑦 → 𝐶 = 𝐷)
cbvixpvw2.2	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvixpvw2	⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑦 ∈ 𝐵 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.

Step	Hyp	Ref	Expression
1		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
2		cbvixpvw2.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
3	1, 2	eleq12d 2834	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	3	cbvabv 2811	. . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐵}
5	4	fneq2i 6664	. . . 4 ⊢ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵})
6		fveq2 6904	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡‘𝑥) = (𝑡‘𝑦))
7		cbvixpvw2.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
8	6, 7	eleq12d 2834	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑡‘𝑥) ∈ 𝐶 ↔ (𝑡‘𝑦) ∈ 𝐷))
9	2, 8	cbvralvw2 36205	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷)
10	5, 9	anbi12i 628	. . 3 ⊢ ((𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶) ↔ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵} ∧ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷))
11	10	abbii 2808	. 2 ⊢ {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶)} = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵} ∧ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷)}
12		df-ixp 8934	. 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶)}
13		df-ixp 8934	. 2 ⊢ X𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵} ∧ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷)}
14	11, 12, 13	3eqtr4i 2774	1 ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑦 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  ∀wral 3060   Fn wfn 6554  ‘cfv 6559  Xcixp 8933

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-iota 6512  df-fn 6562  df-fv 6567  df-ixp 8934

This theorem is referenced by: (None)