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Theorem mpt2eq123i 6140
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 6139 . 2  |-  (  T. 
->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
87trud 1333 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F )
Colors of variables: wff set class
Syntax hints:    T. wtru 1326    = wceq 1653    e. cmpt2 6086
This theorem is referenced by:  ofmres  6346  seqval  11339  oppgtmd  18132  sdc  26462  mendvscafval  27489  tgrpset  31616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-oprab 6088  df-mpt2 6089
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