Theorem cbvmptdavw2 36800

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

Hypotheses

Ref	Expression
cbvmptdavw2.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
cbvmptdavw2.2	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvmptdavw2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐷))

Distinct variable groups:   𝜑,𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbvmptdavw2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2846	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3		cbvmptdavw2.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)
4	3	eleq2d 2849	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
5	2, 4	bitrd 282	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
6		cbvmptdavw2.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)
7	6	eqeq2d 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 = 𝐶 ↔ 𝑡 = 𝐷))
8	5, 7	anbi12d 643	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)))
9	8	cbvopab1davw 36776	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶)} = {⟨𝑦, 𝑡⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)})
10		df-mpt 5193	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶)}
11		df-mpt 5193	. 2 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {⟨𝑦, 𝑡⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)}
12	9, 10, 11	3eqtr4g 2823	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {copab 5173   ↦ cmpt 5192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5174  df-mpt 5193

This theorem is referenced by: (None)