Theorem cbvmpodavw2 37080

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

Hypotheses

Ref	Expression
cbvmpodavw2.1	⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐸 = 𝐹)
cbvmpodavw2.2	⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
cbvmpodavw2.3	⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvmpodavw2	⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹))

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

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

Proof of Theorem cbvmpodavw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
2		cbvmpodavw2.3	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐴 = 𝐵)
3	1, 2	eleq12d 2855	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
4		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
5		cbvmpodavw2.2	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
6	4, 5	eleq12d 2855	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷))
7	3, 6	anbi12d 644	. . . 4 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷)))
8		cbvmpodavw2.1	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐸 = 𝐹)
9	8	eqeq2d 2772	. . . 4 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑡 = 𝐸 ↔ 𝑡 = 𝐹))
10	7, 9	anbi12d 644	. . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹)))
11	10	cbvoprab12davw 37064	. 2 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)} = {⟨⟨𝑧, 𝑤⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹)})
12		df-mpo 7425	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)}
13		df-mpo 7425	. 2 ⊢ (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹) = {⟨⟨𝑧, 𝑤⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹)}
14	11, 12, 13	3eqtr4g 2821	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑤 ∈ 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)