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Theorem mpteq2ia 3885
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 2083 . . 3 𝐴 = 𝐴
21ax-gen 1379 . 2 𝑥 𝐴 = 𝐴
3 mpteq2ia.1 . . 3 (𝑥𝐴𝐵 = 𝐶)
43rgen 2422 . 2 𝑥𝐴 𝐵 = 𝐶
5 mpteq12f 3879 . 2 ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
62, 4, 5mp2an 417 1 (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283   = wceq 1285  wcel 1434  wral 2353  cmpt 3860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-ral 2358  df-opab 3861  df-mpt 3862
This theorem is referenced by:  mpteq2i  3886  feqresmpt  5281  fmptap  5407  offres  5815  cnrecnv  10023
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