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Theorem mpt2eq123i 6342
Description: An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.)
Hypotheses
Ref Expression
mpt2eq123i.1  |-  A  =  D
mpt2eq123i.2  |-  B  =  E
mpt2eq123i.3  |-  C  =  F
Assertion
Ref Expression
mpt2eq123i  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F )

Proof of Theorem mpt2eq123i
StepHypRef Expression
1 mpt2eq123i.1 . . . 4  |-  A  =  D
21a1i 11 . . 3  |-  ( T. 
->  A  =  D
)
3 mpt2eq123i.2 . . . 4  |-  B  =  E
43a1i 11 . . 3  |-  ( T. 
->  B  =  E
)
5 mpt2eq123i.3 . . . 4  |-  C  =  F
65a1i 11 . . 3  |-  ( T. 
->  C  =  F
)
72, 4, 6mpt2eq123dv 6341 . 2  |-  ( T. 
->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
87trud 1388 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   T. wtru 1380    |-> cmpt2 6284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-oprab 6286  df-mpt2 6287
This theorem is referenced by:  ofmres  6777  seqval  12081  dprdvalOLD  16824  oppgtmd  20328  sdc  29838  mendvscafval  30744  tgrpset  35541
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