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Theorem prsdm 29288
 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 5247 . . . 4 dom = dom ((le‘𝐾) ∩ (𝐵 × 𝐵))
32eleq2i 2680 . . 3 (𝑥 ∈ dom 𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4 ordtNEW.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
5 eqid 2610 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
64, 5prsref 16755 . . . . . . . . 9 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → 𝑥(le‘𝐾)𝑥)
7 df-br 4584 . . . . . . . . 9 (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
86, 7sylib 207 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
9 simpr 476 . . . . . . . . 9 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → 𝑥𝐵)
10 opelxpi 5072 . . . . . . . . 9 ((𝑥𝐵𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
119, 10sylancom 698 . . . . . . . 8 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
128, 11elind 3760 . . . . . . 7 ((𝐾 ∈ Preset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
13 vex 3176 . . . . . . . 8 𝑥 ∈ V
14 opeq2 4341 . . . . . . . . 9 (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑥⟩)
1514eleq1d 2672 . . . . . . . 8 (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
1613, 15spcev 3273 . . . . . . 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 3796 . . . . . . . 8 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2019sseli 3564 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
21 opelxp1 5074 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → 𝑥𝐵)
2220, 21syl 17 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2322exlimiv 1845 . . . . 5 (∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2418, 23impbid1 214 . . . 4 (𝐾 ∈ Preset → (𝑥𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
2513eldm2 5244 . . . 4 (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
2624, 25syl6rbbr 278 . . 3 (𝐾 ∈ Preset → (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥𝐵))
273, 26syl5bb 271 . 2 (𝐾 ∈ Preset → (𝑥 ∈ dom 𝑥𝐵))
2827eqrdv 2608 1 (𝐾 ∈ Preset → dom = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ∩ cin 3539  ⟨cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038  ‘cfv 5804  Basecbs 15695  lecple 15775   Preset cpreset 16749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-iota 5768  df-fv 5812  df-preset 16751 This theorem is referenced by:  prsssdm  29291  ordtprsval  29292  ordtprsuni  29293  ordtrestNEW  29295  ordtconlem1  29298
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