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Theorem mptcnv 24076
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 5303 . . 3  |-  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }
2 mptcnv.1 . . . 4  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( y  e.  C  /\  x  =  D ) ) )
32opabbidv 4296 . . 3  |-  ( ph  ->  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( y  e.  C  /\  x  =  D ) } )
41, 3syl5eq 2486 . 2  |-  ( ph  ->  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( y  e.  C  /\  x  =  D ) } )
5 df-mpt 4293 . . 3  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
65cnveqi 5076 . 2  |-  `' ( x  e.  A  |->  B )  =  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
7 df-mpt 4293 . 2  |-  ( y  e.  C  |->  D )  =  { <. y ,  x >.  |  (
y  e.  C  /\  x  =  D ) }
84, 6, 73eqtr4g 2499 1  |-  ( ph  ->  `' ( x  e.  A  |->  B )  =  ( y  e.  C  |->  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1727   {copab 4290    e. cmpt 4291   `'ccnv 4906
This theorem is referenced by:  ballotlemrinv  24822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-mpt 4293  df-xp 4913  df-rel 4914  df-cnv 4915
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