Theorem cbvmpo1vw2 36603

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

Hypotheses

Ref	Expression
cbvmpo1vw2.1	⊢ (𝑥 = 𝑧 → 𝐸 = 𝐹)
cbvmpo1vw2.2	⊢ (𝑥 = 𝑧 → 𝐶 = 𝐷)
cbvmpo1vw2.3	⊢ (𝑥 = 𝑧 → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvmpo1vw2	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹)

Distinct variable groups:   𝑥,𝑦,𝑧   𝑧,𝐴   𝑥,𝐵   𝑧,𝐶   𝑥,𝐷   𝑧,𝐸   𝑥,𝐹

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑦,𝑧)   𝐶(𝑥,𝑦)   𝐷(𝑦,𝑧)   𝐸(𝑥,𝑦)   𝐹(𝑦,𝑧)

Proof of Theorem cbvmpo1vw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
2		cbvmpo1vw2.3	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐴 = 𝐵)
3	1, 2	eleq12d 2856	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
4		cbvmpo1vw2.2	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐷)
5	4	eleq2d 2848	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
6	3, 5	anbi12d 641	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
7		cbvmpo1vw2.1	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐸 = 𝐹)
8	7	eqeq2d 2773	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑡 = 𝐸 ↔ 𝑡 = 𝐹))
9	6, 8	anbi12d 641	. . 3 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)))
10	9	cbvoprab1vw 36597	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)} = {⟨⟨𝑧, 𝑦⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)}
11		df-mpo 7401	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)}
12		df-mpo 7401	. 2 ⊢ (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹) = {⟨⟨𝑧, 𝑦⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)}
13	10, 11, 12	3eqtr4i 2795	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {coprab 7397   ∈ cmpo 7398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-oprab 7400  df-mpo 7401

This theorem is referenced by: (None)