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Theorem psrn 17569
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 17567 . . 3 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
3 eqtr3 2848 . . 3 ((dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅) → dom 𝑅 = ran 𝑅)
42, 3syl 17 . 2 (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
51, 4syl5eq 2873 1 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164   cuni 4660  dom cdm 5346  ran crn 5347  PosetRelcps 17558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ps 17560
This theorem is referenced by:  cnvtsr  17582  ordtbas2  21373  ordtcnv  21383  ordtrest2  21386
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