Theorem prsdm 29960

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 5325	. . . 4 ⊢ dom ≤ = dom ((le‘𝐾) ∩ (𝐵 × 𝐵))
3	2	eleq2i 2693	. . 3 ⊢ (𝑥 ∈ dom ≤ ↔ 𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)))
4		ordtNEW.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
5		eqid 2622	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
6	4, 5	prsref 16932	. . . . . . . . 9 ⊢ ((𝐾 ∈ Preset ∧ 𝑥 ∈ 𝐵) → 𝑥(le‘𝐾)𝑥)
7		df-br 4654	. . . . . . . . 9 ⊢ (𝑥(le‘𝐾)𝑥 ↔ ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
8	6, 7	sylib 208	. . . . . . . 8 ⊢ ((𝐾 ∈ Preset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (le‘𝐾))
9		simpr 477	. . . . . . . . 9 ⊢ ((𝐾 ∈ Preset ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
10		opelxpi 5148	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
11	9, 10	sylancom 701	. . . . . . . 8 ⊢ ((𝐾 ∈ Preset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ (𝐵 × 𝐵))
12	8, 11	elind 3798	. . . . . . 7 ⊢ ((𝐾 ∈ Preset ∧ 𝑥 ∈ 𝐵) → ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
13		vex 3203	. . . . . . . 8 ⊢ 𝑥 ∈ V
14		opeq2 4403	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑥⟩)
15	14	eleq1d 2686	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ⟨𝑥, 𝑥⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
16	13, 15	spcev 3300	. . . . . . 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 450	. . . . 5 ⊢ (𝐾 ∈ Preset → (𝑥 ∈ 𝐵 → ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
19		inss2 3834	. . . . . . . 8 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
20	19	sseli 3599	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
21		opelxp1 5150	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → 𝑥 ∈ 𝐵)
22	20, 21	syl 17	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
23	22	exlimiv 1858	. . . . 5 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)) → 𝑥 ∈ 𝐵)
24	18, 23	impbid1 215	. . . 4 ⊢ (𝐾 ∈ Preset → (𝑥 ∈ 𝐵 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵))))
25	13	eldm2 5322	. . . 4 ⊢ (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ((le‘𝐾) ∩ (𝐵 × 𝐵)))
26	24, 25	syl6rbbr 279	. . 3 ⊢ (𝐾 ∈ Preset → (𝑥 ∈ dom ((le‘𝐾) ∩ (𝐵 × 𝐵)) ↔ 𝑥 ∈ 𝐵))
27	3, 26	syl5bb 272	. 2 ⊢ (𝐾 ∈ Preset → (𝑥 ∈ dom ≤ ↔ 𝑥 ∈ 𝐵))
28	27	eqrdv 2620	1 ⊢ (𝐾 ∈ Preset → dom ≤ = 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ∩ cin 3573  ⟨cop 4183   class class class wbr 4653   × cxp 5112  dom cdm 5114  ‘cfv 5888  Basecbs 15857  lecple 15948   Preset cpreset 16926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-dm 5124  df-iota 5851  df-fv 5896  df-preset 16928

This theorem is referenced by:  prsssdm  29963  ordtprsval  29964  ordtprsuni  29965  ordtrestNEW  29967  ordtconnlem1  29970