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Theorem psrn 18081
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 18079 . . 3 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
3 eqtr3 2763 . . 3 ((dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅) → dom 𝑅 = ran 𝑅)
42, 3syl 17 . 2 (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
51, 4syl5eq 2790 1 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110   cuni 4819  dom cdm 5551  ran crn 5552  PosetRelcps 18070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ps 18072
This theorem is referenced by:  cnvtsr  18094  ordtbas2  22088  ordtcnv  22098  ordtrest2  22101
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