Theorem mpteq1df 45279

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 2732	. 2 ⊢ (𝜑 → 𝐶 = 𝐶)
4	1, 2, 3	mpteq12df 5175	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   = wceq 1541  Ⅎwnf 1784   ↦ cmpt 5172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-opab 5154  df-mpt 5173

This theorem is referenced by:  smfliminflem  46874