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Theorem mpt2eq12 6066
Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpt2eq12  |-  ( ( A  =  C  /\  B  =  D )  ->  ( x  e.  A ,  y  e.  B  |->  E )  =  ( x  e.  C , 
y  e.  D  |->  E ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y
Allowed substitution hints:    E( x, y)

Proof of Theorem mpt2eq12
StepHypRef Expression
1 eqid 2380 . . . . 5  |-  E  =  E
21rgenw 2709 . . . 4  |-  A. y  e.  B  E  =  E
32jctr 527 . . 3  |-  ( B  =  D  ->  ( B  =  D  /\  A. y  e.  B  E  =  E ) )
43ralrimivw 2726 . 2  |-  ( B  =  D  ->  A. x  e.  A  ( B  =  D  /\  A. y  e.  B  E  =  E ) )
5 mpt2eq123 6065 . 2  |-  ( ( A  =  C  /\  A. x  e.  A  ( B  =  D  /\  A. y  e.  B  E  =  E ) )  -> 
( x  e.  A ,  y  e.  B  |->  E )  =  ( x  e.  C , 
y  e.  D  |->  E ) )
64, 5sylan2 461 1  |-  ( ( A  =  C  /\  B  =  D )  ->  ( x  e.  A ,  y  e.  B  |->  E )  =  ( x  e.  C , 
y  e.  D  |->  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649   A.wral 2642    e. cmpt2 6015
This theorem is referenced by:  dffi3  7364  cantnfres  7559  xpsval  13717  homffval  13837  comfffval  13844  monpropd  13883  natfval  14063  plusffval  14622  grpsubfval  14767  grpsubpropd2  14810  lsmvalx  15193  oppglsm  15196  lsmpropd  15229  dvrfval  15709  scaffval  15888  psrmulr  16368  psrplusgpropd  16549  ipffval  16795  txval  17510  cnmptk1p  17631  cnmptk2  17632  xpstopnlem1  17755  pcofval  18899  qqhval2  24158  mendplusgfval  27155  mendmulrfval  27157  mendvscafval  27160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-oprab 6017  df-mpt2 6018
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