Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dranfldrefc Unicode version

Theorem dranfldrefc 25059
Description: The domain and range of a reflexive class are equal. (Contributed by FL, 29-Dec-2011.)
Assertion
Ref Expression
dranfldrefc  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  ->  dom  R  =  ran  R
)
Distinct variable group:    x, R

Proof of Theorem dranfldrefc
StepHypRef Expression
1 domfldrefc 25057 . 2  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  ->  dom  R  =  ( dom 
R  u.  ran  R
) )
2 ranfldrefc 25058 . 2  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  ->  ran  R  =  ( dom 
R  u.  ran  R
) )
31, 2eqtr4d 2318 1  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  ->  dom  R  =  ran  R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   A.wral 2543    u. cun 3150   class class class wbr 4023   dom cdm 4689   ran crn 4690
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-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-cnv 4697  df-dm 4699  df-rn 4700
  Copyright terms: Public domain W3C validator