Theorem prsdm 34249

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.

Step	Hyp	Ref	Expression
1		ordtNEW.l	. . . . 5 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2	1	dmeqi 5895	. . . 4 ⊢ dom ≤ = dom ((le‘𝐾) ∩ (𝐵 × 𝐵))
3	2	eleq2i 2861	. . 3 ⊢ (𝑥 ∈ dom ≤ ↔ 𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
5	4	eldm2 5892	. . . 4 ⊢ (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
6		ordtNEW.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
7		eqid 2769	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
8	6, 7	prsref 18354	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → 𝑥(le‘𝐾)𝑥)
9		df-br 5114	. . . . . . . . 9 ⊢ (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
10	8, 9	sylib 221	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
11		simpr 489	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
12	11, 11	opelxpd 5701	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
13	10, 12	elind 4161	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
14		opeq2 4843	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑥⟩)
15	14	eleq1d 2854	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
16	4, 15	spcev 3574	. . . . . . 7 ⊢ (⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
17	13, 16	syl 18	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ 𝐵) → ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
18	17	ex 417	. . . . 5 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 → ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
19		elinel2 4163	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
20		opelxp1 5704	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → 𝑥 ∈ 𝐵)
21	19, 20	syl 18	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
22	21	exlimiv 1957	. . . . 5 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
23	18, 22	impbid1 228	. . . 4 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
24	5, 23	bitr4id 293	. . 3 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥 ∈ 𝐵))
25	3, 24	bitrid 286	. 2 ⊢ (𝐾 ∈ Proset → (𝑥 ∈ dom ≤ ↔ 𝑥 ∈ 𝐵))
26	25	eqrdv 2767	1 ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ∩ cin 3912  ⟨cop 4600   class class class wbr 5113   × cxp 5660  dom cdm 5662  ‘cfv 6537  Basecbs 17269  lecple 17317   Proset cproset 18348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-dm 5672  df-iota 6493  df-fv 6545  df-proset 18350

This theorem is referenced by:  prsssdm  34252  ordtprsval  34253  ordtprsuni  34254  ordtrestNEW  34256  ordtconnlem1  34259