Theorem cbvmpo1davw2 36334

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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
2		cbvmpo1davw2.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 𝐴 = 𝐵)
3	1, 2	eleq12d 2825	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵))
4		cbvmpo1davw2.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 𝐶 = 𝐷)
5	4	eleq2d 2817	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝑦 ∈ 𝐶 ↔ 𝑦 ∈ 𝐷))
6	3, 5	anbi12d 632	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)))
7		cbvmpo1davw2.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → 𝐸 = 𝐹)
8	7	eqeq2d 2742	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝑡 = 𝐸 ↔ 𝑡 = 𝐹))
9	6, 8	anbi12d 632	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)))
10	9	cbvoprab1davw 36313	. 2 ⊢ (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)} = {⟨⟨𝑧, 𝑦⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)})
11		df-mpo 7351	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑡⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)}
12		df-mpo 7351	. 2 ⊢ (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹) = {⟨⟨𝑧, 𝑦⟩, 𝑡⟩ ∣ ((𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷) ∧ 𝑡 = 𝐹)}
13	10, 11, 12	3eqtr4g 2791	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {coprab 7347   ∈ cmpo 7348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-oprab 7350  df-mpo 7351

This theorem is referenced by: (None)