Theorem mpteq2ia 4115

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 2193	. . 3 ⊢ 𝐴 = 𝐴
2	1	ax-gen 1460	. 2 ⊢ ∀𝑥 𝐴 = 𝐴
3		mpteq2ia.1	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)
4	3	rgen 2547	. 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶
5		mpteq12f 4109	. 2 ⊢ ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶))
6	2, 4, 5	mp2an 426	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4  ∀wal 1362   = wceq 1364   ∈ wcel 2164  ∀wral 2472   ↦ cmpt 4090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-opab 4091  df-mpt 4092

This theorem is referenced by:  mpteq2i  4116  feqresmpt  5611  elfvmptrab  5653  fmptap  5748  offres  6187  cnrecnv  11054  ege2le3  11814  eirraplem  11920  cnmpt1st  14456  cnmpt2nd  14457  expcncf  14763  dvexp  14860  dveflem  14872  dvef  14873  elply2  14881  plyid  14892