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Theorem cnven 7385
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 461 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  A  e.  V )
2 cnvexg 6524 . . 3  |-  ( A  e.  V  ->  `' A  e.  _V )
32adantl 466 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  `' A  e.  _V )
4 cnvf1o 6671 . . 3  |-  ( Rel 
A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
54adantr 465 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  (
x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
6 f1oen2g 7326 . 2  |-  ( ( A  e.  V  /\  `' A  e.  _V  /\  ( x  e.  A  |-> 
U. `' { x } ) : A -1-1-onto-> `' A )  ->  A  ~~  `' A )
71, 3, 5, 6syl3anc 1218 1  |-  ( ( Rel  A  /\  A  e.  V )  ->  A  ~~  `' A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   _Vcvv 2972   {csn 3877   U.cuni 4091   class class class wbr 4292    e. cmpt 4350   `'ccnv 4839   Rel wrel 4845   -1-1-onto->wf1o 5417    ~~ cen 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-1st 6577  df-2nd 6578  df-en 7311
This theorem is referenced by:  cnvfi  7595  lgsquadlem3  22695  cnvct  26015
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