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Theorem psrn 17149
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 ∈ PosetRel → 𝑋 = ran 𝑅)

Proof of Theorem psrn
StepHypRef Expression
1 psref.1 . 2 𝑋 = dom 𝑅
2 psdmrn 17147 . . 3 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
3 eqtr3 2642 . . 3 ((dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅) → dom 𝑅 = ran 𝑅)
42, 3syl 17 . 2 (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
51, 4syl5eq 2667 1 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987   cuni 4409  dom cdm 5084  ran crn 5085  PosetRelcps 17138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ps 17140
This theorem is referenced by:  cnvtsr  17162  ordtbas2  20935  ordtcnv  20945  ordtrest2  20948
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