Theorem cbvmpo1vw2 37032

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 23	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
2		cbvmpo1vw2.3	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐴 = 𝐵)
3	1, 2	eleq12d 2855	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
4		cbvmpo1vw2.2	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐷)
5	4	eleq2d 2847	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
6	3, 5	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
7		cbvmpo1vw2.1	. . . . 5 ⊢ (𝑥 = 𝑧 → 𝐸 = 𝐹)
8	7	eqeq2d 2772	. . . 4 ⊢ (𝑥 = 𝑧 → (𝑡 = 𝐸 ↔ 𝑡 = 𝐹))
9	6, 8	anbi12d 644	. . 3 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)))
10	9	cbvoprab1vw 37026	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)} = {⟨⟨𝑧, 𝑦⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)}
11		df-mpo 7425	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)}
12		df-mpo 7425	. 2 ⊢ (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹) = {⟨⟨𝑧, 𝑦⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)}
13	10, 11, 12	3eqtr4i 2794	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {coprab 7421   ∈ cmpo 7422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-oprab 7424  df-mpo 7425

This theorem is used by: (None)