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Theorem domrngref 25163
Description: Domain and range of a reflexive relation are equal. (Contributed by FL, 6-Oct-2008.)
Assertion
Ref Expression
domrngref  |-  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  ->  dom  R  =  ran  R
)
Distinct variable group:    x, R

Proof of Theorem domrngref
StepHypRef Expression
1 df-ral 2561 . . . 4  |-  ( A. x  e.  U. U. R x R x  <->  A. x
( x  e.  U. U. R  ->  x R x ) )
2 relfld 5214 . . . . . . . . . 10  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
32eqcomd 2301 . . . . . . . . 9  |-  ( Rel 
R  ->  ( dom  R  u.  ran  R )  =  U. U. R
)
43eleq2d 2363 . . . . . . . 8  |-  ( Rel 
R  ->  ( x  e.  ( dom  R  u.  ran  R )  <->  x  e.  U.
U. R ) )
54biimpd 198 . . . . . . 7  |-  ( Rel 
R  ->  ( x  e.  ( dom  R  u.  ran  R )  ->  x  e.  U. U. R ) )
65imim1d 69 . . . . . 6  |-  ( Rel 
R  ->  ( (
x  e.  U. U. R  ->  x R x )  ->  ( x  e.  ( dom  R  u.  ran  R )  ->  x R x ) ) )
7 elun 3329 . . . . . . . 8  |-  ( x  e.  ( dom  R  u.  ran  R )  <->  ( x  e.  dom  R  \/  x  e.  ran  R ) )
87imbi1i 315 . . . . . . 7  |-  ( ( x  e.  ( dom 
R  u.  ran  R
)  ->  x R x )  <->  ( (
x  e.  dom  R  \/  x  e.  ran  R )  ->  x R x ) )
9 jaob 758 . . . . . . . 8  |-  ( ( ( x  e.  dom  R  \/  x  e.  ran  R )  ->  x R x )  <->  ( (
x  e.  dom  R  ->  x R x )  /\  ( x  e. 
ran  R  ->  x R x ) ) )
10 vex 2804 . . . . . . . . . . . 12  |-  x  e. 
_V
1110, 10brelrn 4925 . . . . . . . . . . 11  |-  ( x R x  ->  x  e.  ran  R )
1211imim2i 13 . . . . . . . . . 10  |-  ( ( x  e.  dom  R  ->  x R x )  ->  ( x  e. 
dom  R  ->  x  e. 
ran  R ) )
1312adantr 451 . . . . . . . . 9  |-  ( ( ( x  e.  dom  R  ->  x R x )  /\  ( x  e.  ran  R  ->  x R x ) )  ->  ( x  e. 
dom  R  ->  x  e. 
ran  R ) )
1410, 10breldm 4899 . . . . . . . . . . 11  |-  ( x R x  ->  x  e.  dom  R )
1514imim2i 13 . . . . . . . . . 10  |-  ( ( x  e.  ran  R  ->  x R x )  ->  ( x  e. 
ran  R  ->  x  e. 
dom  R ) )
1615adantl 452 . . . . . . . . 9  |-  ( ( ( x  e.  dom  R  ->  x R x )  /\  ( x  e.  ran  R  ->  x R x ) )  ->  ( x  e. 
ran  R  ->  x  e. 
dom  R ) )
1713, 16impbid 183 . . . . . . . 8  |-  ( ( ( x  e.  dom  R  ->  x R x )  /\  ( x  e.  ran  R  ->  x R x ) )  ->  ( x  e. 
dom  R  <->  x  e.  ran  R ) )
189, 17sylbi 187 . . . . . . 7  |-  ( ( ( x  e.  dom  R  \/  x  e.  ran  R )  ->  x R x )  ->  (
x  e.  dom  R  <->  x  e.  ran  R ) )
198, 18sylbi 187 . . . . . 6  |-  ( ( x  e.  ( dom 
R  u.  ran  R
)  ->  x R x )  ->  (
x  e.  dom  R  <->  x  e.  ran  R ) )
206, 19syl6 29 . . . . 5  |-  ( Rel 
R  ->  ( (
x  e.  U. U. R  ->  x R x )  ->  ( x  e.  dom  R  <->  x  e.  ran  R ) ) )
2120alimdv 1611 . . . 4  |-  ( Rel 
R  ->  ( A. x ( x  e. 
U. U. R  ->  x R x )  ->  A. x ( x  e. 
dom  R  <->  x  e.  ran  R ) ) )
221, 21syl5bi 208 . . 3  |-  ( Rel 
R  ->  ( A. x  e.  U. U. R x R x  ->  A. x
( x  e.  dom  R  <-> 
x  e.  ran  R
) ) )
2322imp 418 . 2  |-  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  ->  A. x ( x  e. 
dom  R  <->  x  e.  ran  R ) )
24 dfcleq 2290 . 2  |-  ( dom 
R  =  ran  R  <->  A. x ( x  e. 
dom  R  <->  x  e.  ran  R ) )
2523, 24sylibr 203 1  |-  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  ->  dom  R  =  ran  R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   A.wral 2556    u. cun 3163   U.cuni 3843   class class class wbr 4039   dom cdm 4705   ran crn 4706   Rel wrel 4710
This theorem is referenced by:  domfldref  25164  preoran2  25333  dfps2  25392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716
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