Theorem mpteq12da 39965

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 2238	. 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶)
4		mpteq12da.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
5	4	ex 397	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐷))
6	1, 5	ralrimi 3106	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
7		mpteq12f 4866	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
8	3, 6, 7	syl2anc 573	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 382  ∀wal 1629   = wceq 1631  Ⅎwnf 1856   ∈ wcel 2145  ∀wral 3061   ↦ cmpt 4864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ral 3066  df-opab 4848  df-mpt 4865

This theorem is referenced by:  smflimmpt  41531  smflimsupmpt  41550  smfliminfmpt  41553