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Theorem psdmrn 14640
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 3511 . . . . 5  |-  dom  R  C_  ( dom  R  u.  ran  R )
2 dmrnssfld 5130 . . . . 5  |-  ( dom 
R  u.  ran  R
)  C_  U. U. R
31, 2sstri 3358 . . . 4  |-  dom  R  C_ 
U. U. R
43a1i 11 . . 3  |-  ( R  e.  PosetRel  ->  dom  R  C_  U. U. R )
5 pslem 14639 . . . . . 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 971 . . . . 5  |-  ( R  e.  PosetRel  ->  ( x  e. 
U. U. R  ->  x R x ) )
7 vex 2960 . . . . . 6  |-  x  e. 
_V
87, 7breldm 5075 . . . . 5  |-  ( x R x  ->  x  e.  dom  R )
96, 8syl6 32 . . . 4  |-  ( R  e.  PosetRel  ->  ( x  e. 
U. U. R  ->  x  e.  dom  R ) )
109ssrdv 3355 . . 3  |-  ( R  e.  PosetRel  ->  U. U. R  C_  dom  R )
114, 10eqssd 3366 . 2  |-  ( R  e.  PosetRel  ->  dom  R  =  U. U. R )
12 ssun2 3512 . . . . 5  |-  ran  R  C_  ( dom  R  u.  ran  R )
1312, 2sstri 3358 . . . 4  |-  ran  R  C_ 
U. U. R
1413a1i 11 . . 3  |-  ( R  e.  PosetRel  ->  ran  R  C_  U. U. R )
157, 7brelrn 5101 . . . . 5  |-  ( x R x  ->  x  e.  ran  R )
166, 15syl6 32 . . . 4  |-  ( R  e.  PosetRel  ->  ( x  e. 
U. U. R  ->  x  e.  ran  R ) )
1716ssrdv 3355 . . 3  |-  ( R  e.  PosetRel  ->  U. U. R  C_  ran  R )
1814, 17eqssd 3366 . 2  |-  ( R  e.  PosetRel  ->  ran  R  =  U. U. R )
1911, 18jca 520 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 360    = wceq 1653    e. wcel 1726    u. cun 3319    C_ wss 3321   U.cuni 4016   class class class wbr 4213   dom cdm 4879   ran crn 4880   PosetRelcps 14625
This theorem is referenced by:  psref  14641  psrn  14642  psss  14647  spwval  14658  tsrdir  14684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ps 14630
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