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Theorem mpteq2ia 3870
 Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Hypothesis
Ref Expression
mpteq2ia.1 (𝑥𝐴𝐵 = 𝐶)
Assertion
Ref Expression
mpteq2ia (𝑥𝐴𝐵) = (𝑥𝐴𝐶)

Proof of Theorem mpteq2ia
StepHypRef Expression
1 eqid 2056 . . 3 𝐴 = 𝐴
21ax-gen 1354 . 2 𝑥 𝐴 = 𝐴
3 mpteq2ia.1 . . 3 (𝑥𝐴𝐵 = 𝐶)
43rgen 2391 . 2 𝑥𝐴 𝐵 = 𝐶
5 mpteq12f 3864 . 2 ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
62, 4, 5mp2an 410 1 (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1257   = wceq 1259   ∈ wcel 1409  ∀wral 2323   ↦ cmpt 3845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-opab 3846  df-mpt 3847 This theorem is referenced by:  mpteq2i  3871  feqresmpt  5254  fmptap  5380  offres  5789  cnrecnv  9731
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