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Theorem prsdm 31279
 Description: Domain of the relation of a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
Hypotheses
Ref Expression
ordtNEW.b 𝐵 = (Base‘𝐾)
ordtNEW.l = ((le‘𝐾) ∩ (𝐵 × 𝐵))
Assertion
Ref Expression
prsdm (𝐾 ∈ Proset → dom = 𝐵)

Proof of Theorem prsdm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtNEW.l . . . . 5 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
21dmeqi 5737 . . . 4 dom = dom ((le‘𝐾) ∩ (𝐵 × 𝐵))
32eleq2i 2881 . . 3 (𝑥 ∈ dom 𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4 vex 3444 . . . . 5 𝑥 ∈ V
54eldm2 5734 . . . 4 (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
6 ordtNEW.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
7 eqid 2798 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
86, 7prsref 17536 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → 𝑥(le‘𝐾)𝑥)
9 df-br 5031 . . . . . . . . 9 (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
108, 9sylib 221 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
11 simpr 488 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → 𝑥𝐵)
1211, 11opelxpd 5557 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
1310, 12elind 4121 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
14 opeq2 4765 . . . . . . . . 9 (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑥⟩)
1514eleq1d 2874 . . . . . . . 8 (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
164, 15spcev 3555 . . . . . . 7 (⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
1713, 16syl 17 . . . . . 6 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
1817ex 416 . . . . 5 (𝐾 ∈ Proset → (𝑥𝐵 → ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
19 elinel2 4123 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
20 opelxp1 5560 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → 𝑥𝐵)
2119, 20syl 17 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2221exlimiv 1931 . . . . 5 (∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2318, 22impbid1 228 . . . 4 (𝐾 ∈ Proset → (𝑥𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
245, 23bitr4id 293 . . 3 (𝐾 ∈ Proset → (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥𝐵))
253, 24syl5bb 286 . 2 (𝐾 ∈ Proset → (𝑥 ∈ dom 𝑥𝐵))
2625eqrdv 2796 1 (𝐾 ∈ Proset → dom = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ∩ cin 3880  ⟨cop 4531   class class class wbr 5030   × cxp 5517  dom cdm 5519  ‘cfv 6324  Basecbs 16477  lecple 16566   Proset cproset 17530 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-dm 5529  df-iota 6283  df-fv 6332  df-proset 17532 This theorem is referenced by:  prsssdm  31282  ordtprsval  31283  ordtprsuni  31284  ordtrestNEW  31286  ordtconnlem1  31289
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