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Theorem psrn 14624
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 14622 . . 3  |-  ( R  e.  PosetRel  ->  ( dom  R  =  U. U. R  /\  ran  R  =  U. U. R ) )
3 eqtr3 2449 . . 3  |-  ( ( dom  R  =  U. U. R  /\  ran  R  =  U. U. R )  ->  dom  R  =  ran  R )
42, 3syl 16 . 2  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
51, 4syl5eq 2474 1  |-  ( R  e.  PosetRel  ->  X  =  ran  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   U.cuni 4002   dom cdm 4864   ran crn 4865   PosetRelcps 14607
This theorem is referenced by:  cnvtsr  14637  spwpr4  14646  spwpr4c  14647  ordtbas2  17238  ordtcnv  17248  ordtrest2  17251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pr 4390
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ps 14612
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