Theorem cbvmpovw2 36458

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

Hypotheses

Ref	Expression
cbvmpovw2.1	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐸 = 𝐹)
cbvmpovw2.2	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
cbvmpovw2.3	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvmpovw2	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹)

Distinct variable groups:   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 cbvmpovw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
2		cbvmpovw2.3	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐴 = 𝐵)
3	1, 2	eleq12d 2831	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
4		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
5		cbvmpovw2.2	. . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
6	4, 5	eleq12d 2831	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷))
7	3, 6	anbi12d 633	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷)))
8		cbvmpovw2.1	. . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐸 = 𝐹)
9	8	eqeq2d 2748	. . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑡 = 𝐸 ↔ 𝑡 = 𝐹))
10	7, 9	anbi12d 633	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹)))
11	10	cbvoprab12v 7458	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)} = {⟨⟨𝑧, 𝑤⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹)}
12		df-mpo 7373	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)}
13		df-mpo 7373	. 2 ⊢ (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹) = {⟨⟨𝑧, 𝑤⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹)}
14	11, 12, 13	3eqtr4i 2770	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {coprab 7369   ∈ cmpo 7370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-oprab 7372  df-mpo 7373

This theorem is referenced by: (None)