Theorem mpteq12da 39452

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

Hypotheses

Ref	Expression
mpteq12da.1	⊢ Ⅎ𝑥𝜑
mpteq12da.2	⊢ (𝜑 → 𝐴 = 𝐶)
mpteq12da.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)

Assertion

Ref	Expression
mpteq12da	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Proof of Theorem mpteq12da

Step	Hyp	Ref	Expression
1		mpteq12da.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		mpteq12da.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
3	1, 2	alrimi 2082	. 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶)
4		mpteq12da.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
5	4	ex 450	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐷))
6	1, 5	ralrimi 2957	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
7		mpteq12f 4731	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
8	3, 6, 7	syl2anc 693	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  ∀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:  smflimmpt  41016  smflimsupmpt  41035  smfliminfmpt  41038