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.

Step	Hyp	Ref	Expression
1		ordtNEW.l	. . . . 5 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2	1	dmeqi 5247	. . . 4 ⊢ dom ≤ = dom ((le‘𝐾) ∩ (𝐵 × 𝐵))
3	2	eleq2i 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)
6	4, 5	prsref 16755	. . . . . . . . 9 ⊢ ((𝐾 ∈ Preset ∧ 𝑥 ∈ 𝐵) → 𝑥(le‘𝐾)𝑥)
7		df-br 4584	. . . . . . . . 9 ⊢ (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
8	6, 7	sylib 207	. . . . . . . 8 ⊢ ((𝐾 ∈ Preset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
9		simpr 476	. . . . . . . . 9 ⊢ ((𝐾 ∈ Preset ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
10		opelxpi 5072	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
11	9, 10	sylancom 698	. . . . . . . 8 ⊢ ((𝐾 ∈ Preset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
12	8, 11	elind 3760	. . . . . . 7 ⊢ ((𝐾 ∈ Preset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
13		vex 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
14		opeq2 4341	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑥⟩)
15	14	eleq1d 2672	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
16	13, 15	spcev 3273	. . . . . . 7 ⊢ (⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
17	12, 16	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ Preset ∧ 𝑥 ∈ 𝐵) → ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
18	17	ex 449	. . . . 5 ⊢ (𝐾 ∈ Preset → (𝑥 ∈ 𝐵 → ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
19		inss2 3796	. . . . . . . 8 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
20	19	sseli 3564	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
21		opelxp1 5074	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → 𝑥 ∈ 𝐵)
22	20, 21	syl 17	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
23	22	exlimiv 1845	. . . . 5 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
24	18, 23	impbid1 214	. . . 4 ⊢ (𝐾 ∈ Preset → (𝑥 ∈ 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
25	13	eldm2 5244	. . . 4 ⊢ (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
26	24, 25	syl6rbbr 278	. . 3 ⊢ (𝐾 ∈ Preset → (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥 ∈ 𝐵))
27	3, 26	syl5bb 271	. 2 ⊢ (𝐾 ∈ Preset → (𝑥 ∈ dom ≤ ↔ 𝑥 ∈ 𝐵))
28	27	eqrdv 2608	1 ⊢ (𝐾 ∈ Preset → dom ≤ = 𝐵)

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