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Theorem cnven 7118
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 448 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  A  e.  V )
2 cnvexg 5345 . . 3  |-  ( A  e.  V  ->  `' A  e.  _V )
32adantl 453 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  `' A  e.  _V )
4 cnvf1o 6384 . . 3  |-  ( Rel 
A  ->  ( x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
54adantr 452 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  (
x  e.  A  |->  U. `' { x } ) : A -1-1-onto-> `' A )
6 f1oen2g 7060 . 2  |-  ( ( A  e.  V  /\  `' A  e.  _V  /\  ( x  e.  A  |-> 
U. `' { x } ) : A -1-1-onto-> `' A )  ->  A  ~~  `' A )
71, 3, 5, 6syl3anc 1184 1  |-  ( ( Rel  A  /\  A  e.  V )  ->  A  ~~  `' A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   _Vcvv 2899   {csn 3757   U.cuni 3957   class class class wbr 4153    e. cmpt 4207   `'ccnv 4817   Rel wrel 4823   -1-1-onto->wf1o 5393    ~~ cen 7042
This theorem is referenced by:  cnvfi  7326  lgsquadlem3  21007  cnvct  23948
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-1st 6288  df-2nd 6289  df-en 7046
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