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Theorem mpt2eq3ia 6141
Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Hypothesis
Ref Expression
mpt2eq3ia.1  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  =  D )
Assertion
Ref Expression
mpt2eq3ia  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  D )

Proof of Theorem mpt2eq3ia
StepHypRef Expression
1 mpt2eq3ia.1 . . . 4  |-  ( ( x  e.  A  /\  y  e.  B )  ->  C  =  D )
213adant1 976 . . 3  |-  ( (  T.  /\  x  e.  A  /\  y  e.  B )  ->  C  =  D )
32mpt2eq3dva 6140 . 2  |-  (  T. 
->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  A , 
y  e.  B  |->  D ) )
43trud 1333 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 360    T. wtru 1326    = wceq 1653    e. wcel 1726    e. cmpt2 6085
This theorem is referenced by:  oprab2co  6434  cnfcomlem  7658  cnfcom2  7661  dfioo2  11007  sadcom  12977  comfffval2  13929  oppchomf  13948  symgga  15111  oppglsm  15278  dfrhm2  15823  cnfldsub  16731  cnflddiv  16733  leordtval  17279  xpstopnlem1  17843  divcn  18900  oprpiece1res1  18978  oprpiece1res2  18979  cxpcn  20631  cnnvm  22176  cnre2csqima  24311  mndpluscn  24314  raddcn  24317  mendplusgfval  27472  elovmpt3rab  28096  elovmpt2wrd  28201  elovmptnn0wrd  28202  wwlknprop  28356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-oprab 6087  df-mpt2 6088
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