Theorem cbvixpvw2 36228

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 2823	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	3	cbvabv 2800	. . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐵}
5	4	fneq2i 6618	. . . 4 ⊢ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵})
6		fveq2 6860	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡‘𝑥) = (𝑡‘𝑦))
7		cbvixpvw2.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
8	6, 7	eleq12d 2823	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑡‘𝑥) ∈ 𝐶 ↔ (𝑡‘𝑦) ∈ 𝐷))
9	2, 8	cbvralvw2 36209	. . . 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 2797	. 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 8873	. 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶)}
13		df-ixp 8873	. 2 ⊢ X𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵} ∧ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷)}
14	11, 12, 13	3eqtr4i 2763	1 ⊢ X𝑥 ∈ 𝐴 𝐶 = X𝑦 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045   Fn wfn 6508  ‘cfv 6513  Xcixp 8872

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-iota 6466  df-fn 6516  df-fv 6521  df-ixp 8873

This theorem is referenced by: (None)