Theorem cbvmpo1davw2 37296

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

Hypotheses

Ref	Expression
cbvmpo1davw2.1	⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 𝐸 = 𝐹)
cbvmpo1davw2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 𝐶 = 𝐷)
cbvmpo1davw2.3	⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvmpo1davw2	⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹))

Distinct variable groups:   𝜑,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 cbvmpo1davw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
2		cbvmpo1davw2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 𝐴 = 𝐵)
3	1, 2	eleq12d 2858	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
4		cbvmpo1davw2.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 𝐶 = 𝐷)
5	4	eleq2d 2850	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
6	3, 5	anbi12d 644	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
7		cbvmpo1davw2.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 𝐸 = 𝐹)
8	7	eqeq2d 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝑡 = 𝐸 ↔ 𝑡 = 𝐹))
9	6, 8	anbi12d 644	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)))
10	9	cbvoprab1davw 37275	. 2 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)} = {⟨⟨𝑧, 𝑦⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)})
11		df-mpo 7456	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)}
12		df-mpo 7456	. 2 ⊢ (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹) = {⟨⟨𝑧, 𝑦⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)}
13	10, 11, 12	3eqtr4g 2823	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {coprab 7452   ∈ cmpo 7453

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-oprab 7455  df-mpo 7456

This theorem is used by: (None)