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Theorem mpt2eq12 5596
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 2082 . . . . 5  |-  E  =  E
21rgenw 2419 . . . 4  |-  A. y  e.  B  E  =  E
32jctr 308 . . 3  |-  ( B  =  D  ->  ( B  =  D  /\  A. y  e.  B  E  =  E ) )
43ralrimivw 2436 . 2  |-  ( B  =  D  ->  A. x  e.  A  ( B  =  D  /\  A. y  e.  B  E  =  E ) )
5 mpt2eq123 5595 . 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 280 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 102    = wceq 1285   A.wral 2349    |-> cmpt2 5545
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 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-oprab 5547  df-mpt2 5548
This theorem is referenced by:  iseqeq1  9524  iseqeq4  9527
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