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Theorem psdmrn 18630
Description: The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.)
Assertion
Ref Expression
psdmrn (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))

Proof of Theorem psdmrn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssun1 4187 . . . . 5 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2 dmrnssfld 5986 . . . . 5 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
31, 2sstri 4004 . . . 4 dom 𝑅 𝑅
43a1i 11 . . 3 (𝑅 ∈ PosetRel → dom 𝑅 𝑅)
5 pslem 18629 . . . . . 6 (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 𝑅𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥𝑥𝑅𝑥) → 𝑥 = 𝑥)))
65simp2d 1142 . . . . 5 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥𝑅𝑥))
7 vex 3481 . . . . . 6 𝑥 ∈ V
87, 7breldm 5921 . . . . 5 (𝑥𝑅𝑥𝑥 ∈ dom 𝑅)
96, 8syl6 35 . . . 4 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥 ∈ dom 𝑅))
109ssrdv 4000 . . 3 (𝑅 ∈ PosetRel → 𝑅 ⊆ dom 𝑅)
114, 10eqssd 4012 . 2 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
12 ssun2 4188 . . . . 5 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
1312, 2sstri 4004 . . . 4 ran 𝑅 𝑅
1413a1i 11 . . 3 (𝑅 ∈ PosetRel → ran 𝑅 𝑅)
157, 7brelrn 5955 . . . . 5 (𝑥𝑅𝑥𝑥 ∈ ran 𝑅)
166, 15syl6 35 . . . 4 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥 ∈ ran 𝑅))
1716ssrdv 4000 . . 3 (𝑅 ∈ PosetRel → 𝑅 ⊆ ran 𝑅)
1814, 17eqssd 4012 . 2 (𝑅 ∈ PosetRel → ran 𝑅 = 𝑅)
1911, 18jca 511 1 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  cun 3960  wss 3962   cuni 4911   class class class wbr 5147  dom cdm 5688  ran crn 5689  PosetRelcps 18621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ps 18623
This theorem is referenced by:  psref  18631  psrn  18632  psss  18637  tsrdir  18661
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