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Theorem mpoeq12 5824
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpoeq12  |-  ( ( 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 mpoeq12
StepHypRef Expression
1 eqid 2137 . . . . 5  |-  E  =  E
21rgenw 2485 . . . 4  |-  A. y  e.  B  E  =  E
32jctr 313 . . 3  |-  ( B  =  D  ->  ( B  =  D  /\  A. y  e.  B  E  =  E ) )
43ralrimivw 2504 . 2  |-  ( B  =  D  ->  A. x  e.  A  ( B  =  D  /\  A. y  e.  B  E  =  E ) )
5 mpoeq123 5823 . 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 284 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 103    = wceq 1331   A.wral 2414    e. cmpo 5769
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-oprab 5771  df-mpo 5772
This theorem is referenced by:  seqeq1  10214  txvalex  12412  txval  12413
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