Theorem cbvixpdavw2 36252

Description: Change bound variable and domain in an indexed Cartesian product. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbvixpdavw2.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
cbvixpdavw2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvixpdavw2	⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑦 ∈ 𝐵 𝐷)

Distinct variable groups:   𝜑,𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbvixpdavw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦)
2		cbvixpdavw2.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
3	1, 2	eleq12d 2838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	3	cbvabdavw 36214	. . . . 5 ⊢ (𝜑 → {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑦 ∣ 𝑦 ∈ 𝐵})
5	4	fneq2d 6668	. . . 4 ⊢ (𝜑 → (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ↔ 𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵}))
6		fveq2 6915	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑡‘𝑥) = (𝑡‘𝑦))
7	6	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡‘𝑥) = (𝑡‘𝑦))
8		cbvixpdavw2.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
9	7, 8	eleq12d 2838	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑡‘𝑥) ∈ 𝐶 ↔ (𝑡‘𝑦) ∈ 𝐷))
10	9, 2	cbvraldva2 3356	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷))
11	5, 10	anbi12d 631	. . 3 ⊢ (𝜑 → ((𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶) ↔ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵} ∧ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷)))
12	11	abbidv 2811	. 2 ⊢ (𝜑 → {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶)} = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵} ∧ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷)})
13		df-ixp 8950	. 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑡 ∣ (𝑡 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑡‘𝑥) ∈ 𝐶)}
14		df-ixp 8950	. 2 ⊢ X𝑦 ∈ 𝐵 𝐷 = {𝑡 ∣ (𝑡 Fn {𝑦 ∣ 𝑦 ∈ 𝐵} ∧ ∀𝑦 ∈ 𝐵 (𝑡‘𝑦) ∈ 𝐷)}
15	12, 13, 14	3eqtr4g 2805	1 ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑦 ∈ 𝐵 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067   Fn wfn 6563  ‘cfv 6568  Xcixp 8949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6520  df-fn 6571  df-fv 6576  df-ixp 8950

This theorem is referenced by: (None)