Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fixcnv Unicode version

Theorem fixcnv 24448
Description: The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
fixcnv  |-  Fix A  =  Fix `' A

Proof of Theorem fixcnv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . 4  |-  x  e. 
_V
21, 1brcnv 4864 . . 3  |-  ( x `' A x  <->  x A x )
31elfix 24443 . . 3  |-  ( x  e.  Fix `' A  <->  x `' A x )
41elfix 24443 . . 3  |-  ( x  e.  Fix A  <->  x A x )
52, 3, 43bitr4ri 269 . 2  |-  ( x  e.  Fix A  <->  x  e.  Fix `' A )
65eqriv 2280 1  |-  Fix A  =  Fix `' A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   class class class wbr 4023   `'ccnv 4688   Fixcfix 24378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-fix 24400
  Copyright terms: Public domain W3C validator