Theorem mpteq12df 5109

Description: An equality inference for the maps-to notation. Compare mpteq12dv 5112. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
mpteq12df.1	⊢ Ⅎ𝑥𝜑
mpteq12df.2	⊢ (𝜑 → 𝐴 = 𝐶)
mpteq12df.3	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
mpteq12df	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Proof of Theorem mpteq12df

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpteq12df.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		nfv 1920	. . 3 ⊢ Ⅎ𝑦𝜑
3		mpteq12df.2	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eleq2d 2818	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
5		mpteq12df.3	. . . . 5 ⊢ (𝜑 → 𝐵 = 𝐷)
6	5	eqeq2d 2749	. . . 4 ⊢ (𝜑 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷))
7	4, 6	anbi12d 634	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)))
8	1, 2, 7	opabbid 5092	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)})
9		df-mpt 5108	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
10		df-mpt 5108	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}
11	8, 9, 10	3eqtr4g 2798	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542  Ⅎwnf 1790   ∈ wcel 2113  {copab 5089   ↦ cmpt 5107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-12 2178  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-opab 5090  df-mpt 5108

This theorem is referenced by:  mpteq12dv  5112  esumrnmpt2  31598  exrecfnlem  35162  smflimmpt  43866