Theorem mpteq1df 46009

Description: An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by SN, 11-Nov-2024.)

Hypotheses

Ref	Expression
mpteq1df.1	⊢ Ⅎ𝑥𝜑
mpteq1df.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
mpteq1df	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Proof of Theorem mpteq1df

Step	Hyp	Ref	Expression
1		mpteq1df.1	. 2 ⊢ Ⅎ𝑥𝜑
2		mpteq1df.2	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqidd 2766	. 2 ⊢ (𝜑 → 𝐶 = 𝐶)
4	1, 2, 3	mpteq12df 5197	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Ⅎwnf 1816   ↦ cmpt 5194

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-opab 5176  df-mpt 5195

This theorem is used by:  smfliminflem  47602