Theorem mpteq1df 39443

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 2082	. 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐵)
4		eqid 2622	. . . 4 ⊢ 𝐶 = 𝐶
5	4	rgenw 2924	. . 3 ⊢ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶
6	5	a1i 11	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐶 = 𝐶)
7		mpteq12f 4731	. 2 ⊢ ((∀𝑥 𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))
8	3, 6, 7	syl2anc 693	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4  ∀wal 1481   = wceq 1483  Ⅎwnf 1708  ∀wral 2912   ↦ cmpt 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-opab 4713  df-mpt 4730

This theorem is referenced by:  smfliminflem  41036