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Theorem mpt2eq123i 6257
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 6256 . 2  |-  ( T. 
->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
87trud 1379 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   T. wtru 1371    |-> cmpt2 6201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-oprab 6203  df-mpt2 6204
This theorem is referenced by:  ofmres  6682  seqval  11933  dprdvalOLD  16608  oppgtmd  19799  sdc  28787  mendvscafval  29694  tgrpset  34712
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