Theorem mpteq12df 5188

Description: An equality inference for the maps-to notation. Compare mpteq12dv 5191. (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 486	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
5	1, 2, 4	mpteq12da 5187	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144   ↦ cmpt 5185

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-opab 5167  df-mpt 5186

This theorem is used by:  esumrnmpt2  34750  exrecfnlem  38439  mpteq1df  46384  smflimmpt  47956