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Theorem mpteq2ia 4104
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 2189 . . 3 𝐴 = 𝐴
21ax-gen 1460 . 2 𝑥 𝐴 = 𝐴
3 mpteq2ia.1 . . 3 (𝑥𝐴𝐵 = 𝐶)
43rgen 2543 . 2 𝑥𝐴 𝐵 = 𝐶
5 mpteq12f 4098 . 2 ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → (𝑥𝐴𝐵) = (𝑥𝐴𝐶))
62, 4, 5mp2an 426 1 (𝑥𝐴𝐵) = (𝑥𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362   = wceq 1364  wcel 2160  wral 2468  cmpt 4079
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 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-opab 4080  df-mpt 4081
This theorem is referenced by:  mpteq2i  4105  feqresmpt  5591  elfvmptrab  5632  fmptap  5727  offres  6160  cnrecnv  10951  ege2le3  11711  eirraplem  11816  cnmpt1st  14245  cnmpt2nd  14246  expcncf  14549  dvexp  14632  dveflem  14644  dvef  14645
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