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Theorem cnven 6833
Description: A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
Assertion
Ref Expression
cnven  |-  ( ( Rel  A  /\  A  e.  V )  ->  A  ~~  `' A )

Proof of Theorem cnven
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr 110 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  A  e.  V )
2 cnvexg 5184 . . 3  |-  ( A  e.  V  ->  `' A  e.  _V )
32adantl 277 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  `' A  e.  _V )
4 cnvf1o 6249 . . 3  |-  ( Rel 
A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
54adantr 276 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  (
x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
6 f1oen2g 6780 . 2  |-  ( ( A  e.  V  /\  `' A  e.  _V  /\  ( x  e.  A  |-> 
U. `' { x } ) : A -1-1-onto-> `' A )  ->  A  ~~  `' A )
71, 3, 5, 6syl3anc 1249 1  |-  ( ( Rel  A  /\  A  e.  V )  ->  A  ~~  `' A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   _Vcvv 2752   {csn 3607   U.cuni 3824   class class class wbr 4018    |-> cmpt 4079   `'ccnv 4643   Rel wrel 4649   -1-1-onto->wf1o 5234    ~~ cen 6763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6164  df-2nd 6165  df-en 6766
This theorem is referenced by:  cnvct  6834  relcnvfi  6969
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