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Theorem psref 17818
Description: A poset is reflexive. (Contributed by NM, 13-May-2008.)
Hypothesis
Ref Expression
psref.1 𝑋 = dom 𝑅
Assertion
Ref Expression
psref ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)

Proof of Theorem psref
StepHypRef Expression
1 psref.1 . . . . 5 𝑋 = dom 𝑅
2 psdmrn 17817 . . . . . 6 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
32simpld 498 . . . . 5 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
41, 3syl5eq 2871 . . . 4 (𝑅 ∈ PosetRel → 𝑋 = 𝑅)
54eleq2d 2901 . . 3 (𝑅 ∈ PosetRel → (𝐴𝑋𝐴 𝑅))
6 pslem 17816 . . . 4 (𝑅 ∈ PosetRel → (((𝐴𝑅𝐴𝐴𝑅𝐴) → 𝐴𝑅𝐴) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐴𝐴𝑅𝐴) → 𝐴 = 𝐴)))
76simp2d 1140 . . 3 (𝑅 ∈ PosetRel → (𝐴 𝑅𝐴𝑅𝐴))
85, 7sylbid 243 . 2 (𝑅 ∈ PosetRel → (𝐴𝑋𝐴𝑅𝐴))
98imp 410 1 ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115   cuni 4824   class class class wbr 5052  dom cdm 5542  ran crn 5543  PosetRelcps 17808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ps 17810
This theorem is referenced by:  psss  17824  psssdm2  17825  ordtt1  21987
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