Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mptcnv Unicode version

Theorem mptcnv 23027
Description: The converse of a mapping function. (Contributed by Thierry Arnoux, 16-Jan-2017.)
Hypothesis
Ref Expression
mptcnv.1  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( y  e.  C  /\  x  =  D ) ) )
Assertion
Ref Expression
mptcnv  |-  ( ph  ->  `' ( x  e.  A  |->  B )  =  ( y  e.  C  |->  D ) )
Distinct variable groups:    x, y, ph    x, C    x, D    y, A    y, B
Allowed substitution hints:    A( x)    B( x)    C( y)    D( y)

Proof of Theorem mptcnv
StepHypRef Expression
1 cnvopab 5083 . . . 4  |-  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }
21a1i 10 . . 3  |-  ( ph  ->  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) } )
3 mptcnv.1 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( y  e.  C  /\  x  =  D ) ) )
43opabbidv 4082 . . 3  |-  ( ph  ->  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( y  e.  C  /\  x  =  D ) } )
52, 4eqtrd 2315 . 2  |-  ( ph  ->  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( y  e.  C  /\  x  =  D ) } )
6 df-mpt 4079 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
76cnveqi 4856 . . 3  |-  `' ( x  e.  A  |->  B )  =  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
87a1i 10 . 2  |-  ( ph  ->  `' ( x  e.  A  |->  B )  =  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) } )
9 df-mpt 4079 . . 3  |-  ( y  e.  C  |->  D )  =  { <. y ,  x >.  |  (
y  e.  C  /\  x  =  D ) }
109a1i 10 . 2  |-  ( ph  ->  ( y  e.  C  |->  D )  =  { <. y ,  x >.  |  ( y  e.  C  /\  x  =  D
) } )
115, 8, 103eqtr4d 2325 1  |-  ( ph  ->  `' ( x  e.  A  |->  B )  =  ( y  e.  C  |->  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {copab 4076    e. cmpt 4077   `'ccnv 4688
This theorem is referenced by:  ballotlemrinv  23092
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697
  Copyright terms: Public domain W3C validator