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Theorem prsdm 32486
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 5860 . . . 4 dom = dom ((le‘𝐾) ∩ (𝐵 × 𝐵))
32eleq2i 2829 . . 3 (𝑥 ∈ dom 𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4 vex 3449 . . . . 5 𝑥 ∈ V
54eldm2 5857 . . . 4 (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
6 ordtNEW.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
7 eqid 2736 . . . . . . . . . 10 (le‘𝐾) = (le‘𝐾)
86, 7prsref 18187 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → 𝑥(le‘𝐾)𝑥)
9 df-br 5106 . . . . . . . . 9 (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
108, 9sylib 217 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
11 simpr 485 . . . . . . . . 9 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → 𝑥𝐵)
1211, 11opelxpd 5671 . . . . . . . 8 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
1310, 12elind 4154 . . . . . . 7 ((𝐾 ∈ Proset ∧ 𝑥𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
14 opeq2 4831 . . . . . . . . 9 (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑥⟩)
1514eleq1d 2822 . . . . . . . 8 (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
164, 15spcev 3565 . . . . . . 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 413 . . . . 5 (𝐾 ∈ Proset → (𝑥𝐵 → ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
19 elinel2 4156 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
20 opelxp1 5674 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → 𝑥𝐵)
2119, 20syl 17 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2221exlimiv 1933 . . . . 5 (∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥𝐵)
2318, 22impbid1 224 . . . 4 (𝐾 ∈ Proset → (𝑥𝐵 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
245, 23bitr4id 289 . . 3 (𝐾 ∈ Proset → (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥𝐵))
253, 24bitrid 282 . 2 (𝐾 ∈ Proset → (𝑥 ∈ dom 𝑥𝐵))
2625eqrdv 2734 1 (𝐾 ∈ Proset → dom = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  cin 3909  cop 4592   class class class wbr 5105   × cxp 5631  dom cdm 5633  cfv 6496  Basecbs 17082  lecple 17139   Proset cproset 18181
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-xp 5639  df-dm 5643  df-iota 6448  df-fv 6504  df-proset 18183
This theorem is referenced by:  prsssdm  32489  ordtprsval  32490  ordtprsuni  32491  ordtrestNEW  32493  ordtconnlem1  32496
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