MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2eq123i Structured version   Unicode version

Theorem mpt2eq123i 6144
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 6143 . 2  |-  ( T. 
->  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
87trud 1378 1  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D ,  y  e.  E  |->  F )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369   T. wtru 1370    e. cmpt2 6088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-oprab 6090  df-mpt2 6091
This theorem is referenced by:  ofmres  6568  seqval  11809  dprdvalOLD  16477  oppgtmd  19648  sdc  28611  mendvscafval  29518  pmatcollpw1  30861  tgrpset  34282
  Copyright terms: Public domain W3C validator