Theorem mpteq1df 40373

Description: An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

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	. . 3 ⊢ Ⅎ𝑥𝜑
2		mpteq1df.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	1, 2	alrimi 2199	. 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐵)
4		eqid 2778	. . . 4 ⊢ 𝐶 = 𝐶
5	4	rgenw 3106	. . 3 ⊢ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶
6	5	a1i 11	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 = 𝐶)
7		mpteq12f 4969	. 2 ⊢ ((∀𝑥 𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
8	3, 6, 7	syl2anc 579	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4  ∀wal 1599   = wceq 1601  Ⅎwnf 1827  ∀wral 3090   ↦ cmpt 4967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-ral 3095  df-opab 4951  df-mpt 4968

This theorem is referenced by:  smfliminflem  41977