Theorem mpteq12d 4866

Description: An equality inference for the maps to notation. Compare mpteq12dv 4865. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
mpteq12d.1	⊢ Ⅎ𝑥𝜑
mpteq12d.3	⊢ (𝜑 → 𝐴 = 𝐶)
mpteq12d.4	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
mpteq12d	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Proof of Theorem mpteq12d

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpteq12d.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		nfv 1994	. . 3 ⊢ Ⅎ𝑦𝜑
3		mpteq12d.3	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eleq2d 2835	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
5		mpteq12d.4	. . . . 5 ⊢ (𝜑 → 𝐵 = 𝐷)
6	5	eqeq2d 2780	. . . 4 ⊢ (𝜑 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷))
7	4, 6	anbi12d 608	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
8	1, 2, 7	opabbid 4847	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)})
9		df-mpt 4862	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
10		df-mpt 4862	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}
11	8, 9, 10	3eqtr4g 2829	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630  Ⅎwnf 1855   ∈ wcel 2144  {copab 4844   ↦ cmpt 4861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-opab 4845  df-mpt 4862

This theorem is referenced by:  esumrnmpt2  30464  smflimmpt  41530