MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psdmrn Structured version   Visualization version   GIF version

Theorem psdmrn 17408
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 3919 . . . . 5 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2 dmrnssfld 5539 . . . . 5 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
31, 2sstri 3753 . . . 4 dom 𝑅 𝑅
43a1i 11 . . 3 (𝑅 ∈ PosetRel → dom 𝑅 𝑅)
5 pslem 17407 . . . . . 6 (𝑅 ∈ PosetRel → (((𝑥𝑅𝑥𝑥𝑅𝑥) → 𝑥𝑅𝑥) ∧ (𝑥 𝑅𝑥𝑅𝑥) ∧ ((𝑥𝑅𝑥𝑥𝑅𝑥) → 𝑥 = 𝑥)))
65simp2d 1138 . . . . 5 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥𝑅𝑥))
7 vex 3343 . . . . . 6 𝑥 ∈ V
87, 7breldm 5484 . . . . 5 (𝑥𝑅𝑥𝑥 ∈ dom 𝑅)
96, 8syl6 35 . . . 4 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥 ∈ dom 𝑅))
109ssrdv 3750 . . 3 (𝑅 ∈ PosetRel → 𝑅 ⊆ dom 𝑅)
114, 10eqssd 3761 . 2 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
12 ssun2 3920 . . . . 5 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
1312, 2sstri 3753 . . . 4 ran 𝑅 𝑅
1413a1i 11 . . 3 (𝑅 ∈ PosetRel → ran 𝑅 𝑅)
157, 7brelrn 5511 . . . . 5 (𝑥𝑅𝑥𝑥 ∈ ran 𝑅)
166, 15syl6 35 . . . 4 (𝑅 ∈ PosetRel → (𝑥 𝑅𝑥 ∈ ran 𝑅))
1716ssrdv 3750 . . 3 (𝑅 ∈ PosetRel → 𝑅 ⊆ ran 𝑅)
1814, 17eqssd 3761 . 2 (𝑅 ∈ PosetRel → ran 𝑅 = 𝑅)
1911, 18jca 555 1 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  cun 3713  wss 3715   cuni 4588   class class class wbr 4804  dom cdm 5266  ran crn 5267  PosetRelcps 17399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ps 17401
This theorem is referenced by:  psref  17409  psrn  17410  psss  17415  tsrdir  17439
  Copyright terms: Public domain W3C validator