Theorem cbvmptvw2 36450

Description: Change bound variable and domain in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.)

Hypotheses

Ref	Expression
cbvmptvw2.1	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
cbvmptvw2.2	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvmptvw2	⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 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 cbvmptvw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2820	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2		cbvmptvw2.2	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
3	2	eleq2d 2823	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	1, 3	bitrd 279	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
5		cbvmptvw2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
6	5	eqeq2d 2748	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑡 = 𝐶 ↔ 𝑡 = 𝐷))
7	4, 6	anbi12d 633	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)))
8	7	cbvopab1v 5178	. 2 ⊢ {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶)} = {⟨𝑦, 𝑡⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)}
9		df-mpt 5182	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶)}
10		df-mpt 5182	. 2 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨𝑦, 𝑡⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)}
11	8, 9, 10	3eqtr4i 2770	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {copab 5162   ↦ cmpt 5181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-mpt 5182

This theorem is referenced by: (None)