Theorem cbvmpo2vw2 36644

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

Hypotheses

Ref	Expression
cbvmpo2vw2.1	⊢ (𝑦 = 𝑧 → 𝐸 = 𝐹)
cbvmpo2vw2.2	⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷)
cbvmpo2vw2.3	⊢ (𝑦 = 𝑧 → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvmpo2vw2	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑥 ∈ 𝐵, 𝑧 ∈ 𝐷 ↦ 𝐹)

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

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

Proof of Theorem cbvmpo2vw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvmpo2vw2.3	. . . . . 6 ⊢ (𝑦 = 𝑧 → 𝐴 = 𝐵)
2	1	eleq2d 2855	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3		id 23	. . . . . 6 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
4		cbvmpo2vw2.2	. . . . . 6 ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷)
5	3, 4	eleq12d 2863	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐶 ↔ 𝑧 ∈ 𝐷))
6	2, 5	anbi12d 643	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐷)))
7		cbvmpo2vw2.1	. . . . 5 ⊢ (𝑦 = 𝑧 → 𝐸 = 𝐹)
8	7	eqeq2d 2780	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑡 = 𝐸 ↔ 𝑡 = 𝐹))
9	6, 8	anbi12d 643	. . 3 ⊢ (𝑦 = 𝑧 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐷) ∧ 𝑡 = 𝐹)))
10	9	cbvoprab2vw 36638	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)} = {⟨⟨𝑥, 𝑧⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐷) ∧ 𝑡 = 𝐹)}
11		df-mpo 7416	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)}
12		df-mpo 7416	. 2 ⊢ (𝑥 ∈ 𝐵, 𝑧 ∈ 𝐷 ↦ 𝐹) = {⟨⟨𝑥, 𝑧⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐷) ∧ 𝑡 = 𝐹)}
13	10, 11, 12	3eqtr4i 2802	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑥 ∈ 𝐵, 𝑧 ∈ 𝐷 ↦ 𝐹)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {coprab 7412   ∈ cmpo 7413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-oprab 7415  df-mpo 7416

This theorem is referenced by: (None)