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Theorem mpoeq3ia 5843
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Hypothesis
Ref Expression
mpoeq3ia.1  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  =  D )
Assertion
Ref Expression
mpoeq3ia  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )

Proof of Theorem mpoeq3ia
StepHypRef Expression
1 mpoeq3ia.1 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  =  D )
213adant1 1000 . . 3  |-  ( ( T.  /\  x  e.  A  /\  y  e.  B )  ->  C  =  D )
32mpoeq3dva 5842 . 2  |-  ( T. 
->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A , 
y  e.  B  |->  D ) )
43mptru 1341 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332   T. wtru 1333    e. wcel 1481    e. cmpo 5783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-oprab 5785  df-mpo 5786
This theorem is referenced by:  mpodifsnif  5871  mposnif  5872  oprab2co  6122  genpdf  7339  dfioo2  9786  iseqvalcbv  10260  divcnap  12761
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