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Theorem mpoeq123i 5899
Description: An equality inference for the maps-to notation. (Contributed by NM, 15-Jul-2013.)
Hypotheses
Ref Expression
mpoeq123i.1  |-  A  =  D
mpoeq123i.2  |-  B  =  E
mpoeq123i.3  |-  C  =  F
Assertion
Ref Expression
mpoeq123i  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F )

Proof of Theorem mpoeq123i
StepHypRef Expression
1 mpoeq123i.1 . . . 4  |-  A  =  D
21a1i 9 . . 3  |-  ( T. 
->  A  =  D
)
3 mpoeq123i.2 . . . 4  |-  B  =  E
43a1i 9 . . 3  |-  ( T. 
->  B  =  E
)
5 mpoeq123i.3 . . . 4  |-  C  =  F
65a1i 9 . . 3  |-  ( T. 
->  C  =  F
)
72, 4, 6mpoeq123dv 5898 . 2  |-  ( T. 
->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
87mptru 1351 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F )
Colors of variables: wff set class
Syntax hints:    = wceq 1342   T. wtru 1343    e. cmpo 5841
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-oprab 5843  df-mpo 5844
This theorem is referenced by:  ofmres  6099
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