MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnven Structured version   Unicode version

Theorem cnven 7631
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 6732 . . 3  |-  ( A  e.  V  ->  `' A  e.  _V )
32adantl 466 . 2  |-  ( ( Rel  A  /\  A  e.  V )  ->  `' A  e.  _V )
4 cnvf1o 6885 . . 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 7572 . 2  |-  ( ( A  e.  V  /\  `' A  e.  _V  /\  ( x  e.  A  |-> 
U. `' { x } ) : A -1-1-onto-> `' A )  ->  A  ~~  `' A )
71, 3, 5, 6syl3anc 1232 1  |-  ( ( Rel  A  /\  A  e.  V )  ->  A  ~~  `' A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1844   _Vcvv 3061   {csn 3974   U.cuni 4193   class class class wbr 4397    |-> cmpt 4455   `'ccnv 4824   Rel wrel 4830   -1-1-onto->wf1o 5570    ~~ cen 7553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-1st 6786  df-2nd 6787  df-en 7557
This theorem is referenced by:  cnvfi  7840  lgsquadlem3  24014  cnvct  27997
  Copyright terms: Public domain W3C validator