Theorem mpteq2ia 3954

Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Hypothesis

Ref	Expression
mpteq2ia.1	⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)

Assertion

Ref	Expression
mpteq2ia	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)

Proof of Theorem mpteq2ia

Step	Hyp	Ref	Expression
1		eqid 2100	. . 3 ⊢ 𝐴 = 𝐴
2	1	ax-gen 1393	. 2 ⊢ ∀𝑥 𝐴 = 𝐴
3		mpteq2ia.1	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)
4	3	rgen 2444	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶
5		mpteq12f 3948	. 2 ⊢ ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
6	2, 4, 5	mp2an 420	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4  ∀wal 1297   = wceq 1299   ∈ wcel 1448  ∀wral 2375   ↦ cmpt 3929

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-ral 2380  df-opab 3930  df-mpt 3931

This theorem is referenced by:  mpteq2i  3955  feqresmpt  5407  elfvmptrab  5448  fmptap  5542  offres  5964  cnrecnv  10523  ege2le3  11175  eirraplem  11278  cnmpt1st  12238  cnmpt2nd  12239  expcncf  12504