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Theorem psdmrn 17522
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 3974 . . . . 5 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2 dmrnssfld 5588 . . . . 5 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
31, 2sstri 3807 . . . 4 dom 𝑅 𝑅
43a1i 11 . . 3 (𝑅 ∈ PosetRel → dom 𝑅 𝑅)
5 pslem 17521 . . . . . 6 (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 𝑅𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥𝑥𝑅𝑥) → 𝑥 = 𝑥)))
65simp2d 1174 . . . . 5 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥𝑅𝑥))
7 vex 3388 . . . . . 6 𝑥 ∈ V
87, 7breldm 5532 . . . . 5 (𝑥𝑅𝑥𝑥 ∈ dom 𝑅)
96, 8syl6 35 . . . 4 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥 ∈ dom 𝑅))
109ssrdv 3804 . . 3 (𝑅 ∈ PosetRel → 𝑅 ⊆ dom 𝑅)
114, 10eqssd 3815 . 2 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
12 ssun2 3975 . . . . 5 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
1312, 2sstri 3807 . . . 4 ran 𝑅 𝑅
1413a1i 11 . . 3 (𝑅 ∈ PosetRel → ran 𝑅 𝑅)
157, 7brelrn 5560 . . . . 5 (𝑥𝑅𝑥𝑥 ∈ ran 𝑅)
166, 15syl6 35 . . . 4 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥 ∈ ran 𝑅))
1716ssrdv 3804 . . 3 (𝑅 ∈ PosetRel → 𝑅 ⊆ ran 𝑅)
1814, 17eqssd 3815 . 2 (𝑅 ∈ PosetRel → ran 𝑅 = 𝑅)
1911, 18jca 508 1 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  cun 3767  wss 3769   cuni 4628   class class class wbr 4843  dom cdm 5312  ran crn 5313  PosetRelcps 17513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ps 17515
This theorem is referenced by:  psref  17523  psrn  17524  psss  17529  tsrdir  17553
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