Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mpteq12d Structured version   Unicode version

Theorem mpteq12d 29367
Description: An equality inference for the maps to notation. Compare mpteq12dv 4445. (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 1715 . . 3  |-  F/ y
ph
3 mpteq12d.3 . . . . 5  |-  ( ph  ->  A  =  C )
43eleq2d 2452 . . . 4  |-  ( ph  ->  ( x  e.  A  <->  x  e.  C ) )
5 mpteq12d.4 . . . . 5  |-  ( ph  ->  B  =  D )
65eqeq2d 2396 . . . 4  |-  ( ph  ->  ( y  =  B  <-> 
y  =  D ) )
74, 6anbi12d 708 . . 3  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  C  /\  y  =  D ) ) )
81, 2, 7opabbid 4429 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. x ,  y
>.  |  ( x  e.  C  /\  y  =  D ) } )
9 df-mpt 4427 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
10 df-mpt 4427 . 2  |-  ( x  e.  C  |->  D )  =  { <. x ,  y >.  |  ( x  e.  C  /\  y  =  D ) }
118, 9, 103eqtr4g 2448 1  |-  ( ph  ->  ( x  e.  A  |->  B )  =  ( x  e.  C  |->  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399   F/wnf 1624    e. wcel 1826   {copab 4424    |-> cmpt 4425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-opab 4426  df-mpt 4427
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator