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Theorem psdmrn 17128
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 3754 . . . . 5 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2 dmrnssfld 5344 . . . . 5 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
31, 2sstri 3592 . . . 4 dom 𝑅 𝑅
43a1i 11 . . 3 (𝑅 ∈ PosetRel → dom 𝑅 𝑅)
5 pslem 17127 . . . . . 6 (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 𝑅𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥𝑥𝑅𝑥) → 𝑥 = 𝑥)))
65simp2d 1072 . . . . 5 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥𝑅𝑥))
7 vex 3189 . . . . . 6 𝑥 ∈ V
87, 7breldm 5289 . . . . 5 (𝑥𝑅𝑥𝑥 ∈ dom 𝑅)
96, 8syl6 35 . . . 4 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥 ∈ dom 𝑅))
109ssrdv 3589 . . 3 (𝑅 ∈ PosetRel → 𝑅 ⊆ dom 𝑅)
114, 10eqssd 3600 . 2 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
12 ssun2 3755 . . . . 5 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
1312, 2sstri 3592 . . . 4 ran 𝑅 𝑅
1413a1i 11 . . 3 (𝑅 ∈ PosetRel → ran 𝑅 𝑅)
157, 7brelrn 5316 . . . . 5 (𝑥𝑅𝑥𝑥 ∈ ran 𝑅)
166, 15syl6 35 . . . 4 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥 ∈ ran 𝑅))
1716ssrdv 3589 . . 3 (𝑅 ∈ PosetRel → 𝑅 ⊆ ran 𝑅)
1814, 17eqssd 3600 . 2 (𝑅 ∈ PosetRel → ran 𝑅 = 𝑅)
1911, 18jca 554 1 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cun 3553  wss 3555   cuni 4402   class class class wbr 4613  dom cdm 5074  ran crn 5075  PosetRelcps 17119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ps 17121
This theorem is referenced by:  psref  17129  psrn  17130  psss  17135  tsrdir  17159
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