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Theorem mpoeq123dv 5936
Description: An equality deduction for the maps-to notation. (Contributed by NM, 12-Sep-2011.)
Hypotheses
Ref Expression
mpoeq123dv.1  |-  ( ph  ->  A  =  D )
mpoeq123dv.2  |-  ( ph  ->  B  =  E )
mpoeq123dv.3  |-  ( ph  ->  C  =  F )
Assertion
Ref Expression
mpoeq123dv  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
Distinct variable groups:    ph, x    ph, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)    D( x, y)    E( x, y)    F( x, y)

Proof of Theorem mpoeq123dv
StepHypRef Expression
1 mpoeq123dv.1 . 2  |-  ( ph  ->  A  =  D )
2 mpoeq123dv.2 . . 3  |-  ( ph  ->  B  =  E )
32adantr 276 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  =  E )
4 mpoeq123dv.3 . . 3  |-  ( ph  ->  C  =  F )
54adantr 276 . 2  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  C  =  F )
61, 3, 5mpoeq123dva 5935 1  |-  ( ph  ->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    e. cmpo 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-oprab 5878  df-mpo 5879
This theorem is referenced by:  mpoeq123i  5937  prdsex  12712  plusffvalg  12775  grpsubfvalg  12912  grpsubpropdg  12968  mulgfvalg  12979  mulgpropdg  13018  dvrfvald  13295  blfvalps  13816
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