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Theorem mpteq12d 27605
Description: An equality inference for the maps to notation. Compare mpteq12dv 4391. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
mpteq12d.1  |-  F/ x ph
mpteq12d.3  |-  ( ph  ->  A  =  C )
mpteq12d.4  |-  ( ph  ->  B  =  D )
Assertion
Ref Expression
mpteq12d  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )

Proof of Theorem mpteq12d
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mpteq12d.1 . . 3  |-  F/ x ph
2 nfv 1673 . . 3  |-  F/ y
ph
3 mpteq12d.3 . . . . 5  |-  ( ph  ->  A  =  C )
43eleq2d 2510 . . . 4  |-  ( ph  ->  ( x  e.  A  <->  x  e.  C ) )
5 mpteq12d.4 . . . . 5  |-  ( ph  ->  B  =  D )
65eqeq2d 2454 . . . 4  |-  ( ph  ->  ( y  =  B  <-> 
y  =  D ) )
74, 6anbi12d 710 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  C  /\  y  =  D ) ) )
81, 2, 7opabbid 4375 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. x ,  y
>.  |  ( x  e.  C  /\  y  =  D ) } )
9 df-mpt 4373 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
10 df-mpt 4373 . 2  |-  ( x  e.  C  |->  D )  =  { <. x ,  y >.  |  ( x  e.  C  /\  y  =  D ) }
118, 9, 103eqtr4g 2500 1  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   F/wnf 1589    e. wcel 1756   {copab 4370    e. cmpt 4371
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 2423
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 2430  df-cleq 2436  df-clel 2439  df-opab 4372  df-mpt 4373
This theorem is referenced by: (None)
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