Theorem cbvixpvw2 36279

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 2825	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	3	cbvabv 2801	. . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐵}
5	4	fneq2i 6574	. . . 4 ⊢ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵})
6		fveq2 6817	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡‘𝑥) = (𝑡‘𝑦))
7		cbvixpvw2.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
8	6, 7	eleq12d 2825	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑡‘𝑥) ∈ 𝐶 ↔ (𝑡‘𝑦) ∈ 𝐷))
9	2, 8	cbvralvw2 36260	. . . 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 2798	. 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 8817	. 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶)}
13		df-ixp 8817	. 2 ⊢ X𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵} ∧ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷)}
14	11, 12, 13	3eqtr4i 2764	1 ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑦 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047   Fn wfn 6471  ‘cfv 6476  Xcixp 8816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fn 6479  df-fv 6484  df-ixp 8817

This theorem is referenced by: (None)