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Theorem prsdm 30088
Description: Domain of the relation of a preset. (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 ∈ Preset → 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 5357 . . . 4 dom = dom ((le‘𝐾) ∩ (𝐵 × 𝐵))
32eleq2i 2722 . . 3 (𝑥 ∈ dom 𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4 ordtNEW.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
5 eqid 2651 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
64, 5prsref 16979 . . . . . . . . 9 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → 𝑥(le‘𝐾)𝑥)
7 df-br 4686 . . . . . . . . 9 (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
86, 7sylib 208 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
9 simpr 476 . . . . . . . . 9 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → 𝑥𝐵)
10 opelxpi 5182 . . . . . . . . 9 ((𝑥𝐵𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
119, 10sylancom 702 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
128, 11elind 3831 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
13 vex 3234 . . . . . . . 8 𝑥 ∈ V
14 opeq2 4434 . . . . . . . . 9 (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑥⟩)
1514eleq1d 2715 . . . . . . . 8 (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
1613, 15spcev 3331 . . . . . . 7 (⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
1712, 16syl 17 . . . . . 6 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
1817ex 449 . . . . 5 (𝐾 ∈ Preset → (𝑥𝐵 → ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
19 inss2 3867 . . . . . . . 8 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2019sseli 3632 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
21 opelxp1 5184 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → 𝑥𝐵)
2220, 21syl 17 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2322exlimiv 1898 . . . . 5 (∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2418, 23impbid1 215 . . . 4 (𝐾 ∈ Preset → (𝑥𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
2513eldm2 5354 . . . 4 (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
2624, 25syl6rbbr 279 . . 3 (𝐾 ∈ Preset → (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥𝐵))
273, 26syl5bb 272 . 2 (𝐾 ∈ Preset → (𝑥 ∈ dom 𝑥𝐵))
2827eqrdv 2649 1 (𝐾 ∈ Preset → dom = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  cin 3606  cop 4216   class class class wbr 4685   × cxp 5141  dom cdm 5143  cfv 5926  Basecbs 15904  lecple 15995   Preset cpreset 16973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-dm 5153  df-iota 5889  df-fv 5934  df-preset 16975
This theorem is referenced by:  prsssdm  30091  ordtprsval  30092  ordtprsuni  30093  ordtrestNEW  30095  ordtconnlem1  30098
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