Theorem mpteq12df 5156

Description: An equality inference for the maps-to notation. Compare mpteq12dv 5159. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) (Proof shortened by SN, 11-Nov-2024.)

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

Step	Hyp	Ref	Expression
1		mpteq12df.1	. 2 ⊢ Ⅎ𝑥𝜑
2		mpteq12df.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
3		mpteq12df.3	. . 3 ⊢ (𝜑 → 𝐵 = 𝐷)
4	3	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
5	1, 2, 4	mpteq12da 5155	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119   ↦ cmpt 5153

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 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-opab 5135  df-mpt 5154

This theorem is referenced by:  esumrnmpt2  34252  exrecfnlem  37741  mpteq1df  45680  smflimmpt  47253