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Theorem cnven 6670
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 109 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  A  e.  V )
2 cnvexg 5046 . . 3  |-  ( A  e.  V  ->  `' A  e.  _V )
32adantl 275 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  `' A  e.  _V )
4 cnvf1o 6090 . . 3  |-  ( Rel 
A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
54adantr 274 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  (
x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
6 f1oen2g 6617 . 2  |-  ( ( A  e.  V  /\  `' A  e.  _V  /\  ( x  e.  A  |-> 
U. `' { x } ) : A -1-1-onto-> `' A )  ->  A  ~~  `' A )
71, 3, 5, 6syl3anc 1201 1  |-  ( ( Rel  A  /\  A  e.  V )  ->  A  ~~  `' A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465   _Vcvv 2660   {csn 3497   U.cuni 3706   class class class wbr 3899    |-> cmpt 3959   `'ccnv 4508   Rel wrel 4514   -1-1-onto->wf1o 5092    ~~ cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-en 6603
This theorem is referenced by:  cnvct  6671  relcnvfi  6797
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