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Theorem mpt2eq123i 5911
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 10 . . 3  |-  (  T. 
->  A  =  D
)
3 mpt2eq123i.2 . . . 4  |-  B  =  E
43a1i 10 . . 3  |-  (  T. 
->  B  =  E
)
5 mpt2eq123i.3 . . . 4  |-  C  =  F
65a1i 10 . . 3  |-  (  T. 
->  C  =  F
)
72, 4, 6mpt2eq123dv 5910 . 2  |-  (  T. 
->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
87trud 1314 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 1307    = wceq 1623    e. cmpt2 5860
This theorem is referenced by:  ofmres  6116  seqval  11057  oppgtmd  17780  sdc  26454  mendvscafval  27498  tgrpset  30934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-oprab 5862  df-mpt2 5863
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