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Theorem psdmrn 14316
Description: The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.)
Assertion
Ref Expression
psdmrn  |-  ( R  e.  PosetRel  ->  ( dom  R  =  U. U. R  /\  ran  R  =  U. U. R ) )

Proof of Theorem psdmrn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssun1 3338 . . . . 5  |-  dom  R  C_  ( dom  R  u.  ran  R )
2 dmrnssfld 4938 . . . . 5  |-  ( dom 
R  u.  ran  R
)  C_  U. U. R
31, 2sstri 3188 . . . 4  |-  dom  R  C_ 
U. U. R
43a1i 10 . . 3  |-  ( R  e.  PosetRel  ->  dom  R  C_  U. U. R )
5 pslem 14315 . . . . . 6  |-  ( R  e.  PosetRel  ->  ( ( ( x R x  /\  x R x )  ->  x R x )  /\  ( x  e.  U. U. R  ->  x R x )  /\  ( ( x R x  /\  x R x )  ->  x  =  x )
) )
65simp2d 968 . . . . 5  |-  ( R  e.  PosetRel  ->  ( x  e. 
U. U. R  ->  x R x ) )
7 vex 2791 . . . . . 6  |-  x  e. 
_V
87, 7breldm 4883 . . . . 5  |-  ( x R x  ->  x  e.  dom  R )
96, 8syl6 29 . . . 4  |-  ( R  e.  PosetRel  ->  ( x  e. 
U. U. R  ->  x  e.  dom  R ) )
109ssrdv 3185 . . 3  |-  ( R  e.  PosetRel  ->  U. U. R  C_  dom  R )
114, 10eqssd 3196 . 2  |-  ( R  e.  PosetRel  ->  dom  R  =  U. U. R )
12 ssun2 3339 . . . . 5  |-  ran  R  C_  ( dom  R  u.  ran  R )
1312, 2sstri 3188 . . . 4  |-  ran  R  C_ 
U. U. R
1413a1i 10 . . 3  |-  ( R  e.  PosetRel  ->  ran  R  C_  U. U. R )
157, 7brelrn 4909 . . . . 5  |-  ( x R x  ->  x  e.  ran  R )
166, 15syl6 29 . . . 4  |-  ( R  e.  PosetRel  ->  ( x  e. 
U. U. R  ->  x  e.  ran  R ) )
1716ssrdv 3185 . . 3  |-  ( R  e.  PosetRel  ->  U. U. R  C_  ran  R )
1814, 17eqssd 3196 . 2  |-  ( R  e.  PosetRel  ->  ran  R  =  U. U. R )
1911, 18jca 518 1  |-  ( R  e.  PosetRel  ->  ( dom  R  =  U. U. R  /\  ran  R  =  U. U. R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150    C_ wss 3152   U.cuni 3827   class class class wbr 4023   dom cdm 4689   ran crn 4690   PosetRelcps 14301
This theorem is referenced by:  psref  14317  psrn  14318  psss  14323  spwval  14334  tsrdir  14360
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-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ps 14306
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