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Theorem psdmrn 17807
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 4152 . . . . 5 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2 dmrnssfld 5840 . . . . 5 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
31, 2sstri 3980 . . . 4 dom 𝑅 𝑅
43a1i 11 . . 3 (𝑅 ∈ PosetRel → dom 𝑅 𝑅)
5 pslem 17806 . . . . . 6 (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 𝑅𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥𝑥𝑅𝑥) → 𝑥 = 𝑥)))
65simp2d 1137 . . . . 5 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥𝑅𝑥))
7 vex 3503 . . . . . 6 𝑥 ∈ V
87, 7breldm 5776 . . . . 5 (𝑥𝑅𝑥𝑥 ∈ dom 𝑅)
96, 8syl6 35 . . . 4 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥 ∈ dom 𝑅))
109ssrdv 3977 . . 3 (𝑅 ∈ PosetRel → 𝑅 ⊆ dom 𝑅)
114, 10eqssd 3988 . 2 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
12 ssun2 4153 . . . . 5 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
1312, 2sstri 3980 . . . 4 ran 𝑅 𝑅
1413a1i 11 . . 3 (𝑅 ∈ PosetRel → ran 𝑅 𝑅)
157, 7brelrn 5811 . . . . 5 (𝑥𝑅𝑥𝑥 ∈ ran 𝑅)
166, 15syl6 35 . . . 4 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥 ∈ ran 𝑅))
1716ssrdv 3977 . . 3 (𝑅 ∈ PosetRel → 𝑅 ⊆ ran 𝑅)
1814, 17eqssd 3988 . 2 (𝑅 ∈ PosetRel → ran 𝑅 = 𝑅)
1911, 18jca 512 1 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  cun 3938  wss 3940   cuni 4837   class class class wbr 5063  dom cdm 5554  ran crn 5555  PosetRelcps 17798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ps 17800
This theorem is referenced by:  psref  17808  psrn  17809  psss  17814  tsrdir  17838
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