Theorem mpteq12dva 4732

Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

Ref	Expression
mpteq12dv.1	⊢ (𝜑 → 𝐴 = 𝐶)
mpteq12dva.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)

Assertion

Ref	Expression
mpteq12dva	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem mpteq12dva

Step	Hyp	Ref	Expression
1		mpteq12dv.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
2	1	alrimiv 1855	. 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶)
3		mpteq12dva.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷)
4	3	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷)
5		mpteq12f 4731	. 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))
6	2, 4, 5	syl2anc 693	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ 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:  mpteq12dv  4733  reps  13517  repswccat  13532  cidpropd  16370  monpropd  16397  fucpropd  16637  curfpropd  16873  hofpropd  16907  yonffthlem  16922  ofco2  20257  pmatcollpw3fi1lem1  20591  rrxnm  23179  ushgredgedg  26121  ushgredgedgloop  26123  sgnsv  29727  ofcfval  30160  ccatmulgnn0dir  30619  signstf0  30645  curunc  33391  cncfiooicc  40107  dvcosax  40141  fourierdlem74  40397  fourierdlem75  40398  fourierdlem93  40416  smfsupxr  41022  smflimsuplem8  41033  pfxmpt  41387