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Theorem mptcnv 24006
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 5241 . . 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 4239 . . 3  |-  ( ph  ->  { <. y ,  x >.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( y  e.  C  /\  x  =  D ) } )
41, 3syl5eq 2456 . 2  |-  ( ph  ->  `' { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  =  { <. y ,  x >.  |  ( y  e.  C  /\  x  =  D ) } )
5 df-mpt 4236 . . 3  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
65cnveqi 5014 . 2  |-  `' ( x  e.  A  |->  B )  =  `' { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
7 df-mpt 4236 . 2  |-  ( y  e.  C  |->  D )  =  { <. y ,  x >.  |  (
y  e.  C  /\  x  =  D ) }
84, 6, 73eqtr4g 2469 1  |-  ( ph  ->  `' ( x  e.  A  |->  B )  =  ( y  e.  C  |->  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {copab 4233    e. cmpt 4234   `'ccnv 4844
This theorem is referenced by:  ballotlemrinv  24752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-mpt 4236  df-xp 4851  df-rel 4852  df-cnv 4853
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