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Theorem psrn 14334
Description: The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)
Hypothesis
Ref Expression
psref.1  |-  X  =  dom  R
Assertion
Ref Expression
psrn  |-  ( R  e.  PosetRel  ->  X  =  ran  R )

Proof of Theorem psrn
StepHypRef Expression
1 psref.1 . 2  |-  X  =  dom  R
2 psdmrn 14332 . . 3  |-  ( R  e.  PosetRel  ->  ( dom  R  =  U. U. R  /\  ran  R  =  U. U. R ) )
3 eqtr3 2315 . . 3  |-  ( ( dom  R  =  U. U. R  /\  ran  R  =  U. U. R )  ->  dom  R  =  ran  R )
42, 3syl 15 . 2  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
51, 4syl5eq 2340 1  |-  ( R  e.  PosetRel  ->  X  =  ran  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   U.cuni 3843   dom cdm 4705   ran crn 4706   PosetRelcps 14317
This theorem is referenced by:  cnvtsr  14347  spwpr4  14356  spwpr4c  14357  ordtbas2  16937  ordtcnv  16947  ordtrest2  16950  cnvordtrestixx  23312  xrge0iifhmeo  23333
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-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ps 14322
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