Theorem ordtrestNEW 34311

Description: The subspace topology of an order topology is in general finer than the topology generated by the restricted order, but we do have inclusion in one direction. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)

Hypotheses

Ref	Expression
ordtNEW.b	⊢ 𝐵 = (Base‘𝐾)
ordtNEW.l	⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))

Assertion

Ref	Expression
ordtrestNEW	⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))

Proof of Theorem ordtrestNEW

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		fvex 6894	. . . . . 6 ⊢ (le‘𝐾) ∈ V
3	2	inex1 5286	. . . . 5 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
4	1, 3	eqeltri 2859	. . . 4 ⊢ ≤ ∈ V
5	4	inex1 5286	. . 3 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) ∈ V
6		eqid 2763	. . . 4 ⊢ dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ( ≤ ∩ (𝐴 × 𝐴))
7		eqid 2763	. . . 4 ⊢ ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) = ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥})
8		eqid 2763	. . . 4 ⊢ ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}) = ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
9	6, 7, 8	ordtval 23346	. . 3 ⊢ (( ≤ ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = (topGen‘(fi‘({dom ( ≤ ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}))))))
10	5, 9	mp1i 14	. 2 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = (topGen‘(fi‘({dom ( ≤ ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}))))))
11		ordttop 23357	. . . . . 6 ⊢ ( ≤ ∈ V → (ordTop‘ ≤ ) ∈ Top)
12	4, 11	ax-mp 5	. . . . 5 ⊢ (ordTop‘ ≤ ) ∈ Top
13		ordtNEW.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
14		fvex 6894	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
15	13, 14	eqeltri 2859	. . . . . 6 ⊢ 𝐵 ∈ V
16	15	ssex 5291	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
17		resttop 23317	. . . . 5 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V) → ((ordTop‘ ≤ ) ↾t 𝐴) ∈ Top)
18	12, 16, 17	sylancr 598	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((ordTop‘ ≤ ) ↾t 𝐴) ∈ Top)
19	18	adantl 486	. . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((ordTop‘ ≤ ) ↾t 𝐴) ∈ Top)
20	13	ressprs 33286	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Proset )
21		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴))
22		eqid 2763	. . . . . . . . . 10 ⊢ ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
23	21, 22	prsdm 34304	. . . . . . . . 9 ⊢ ((𝐾 ↾s 𝐴) ∈ Proset → dom ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) = (Base‘(𝐾 ↾s 𝐴)))
24	20, 23	syl 18	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) = (Base‘(𝐾 ↾s 𝐴)))
25		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴)
26	25, 13	ressbas2 17293	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
27		fvex 6894	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐾 ↾s 𝐴)) ∈ V
28	26, 27	eqeltrdi 2871	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
29		eqid 2763	. . . . . . . . . . . . 13 ⊢ (le‘𝐾) = (le‘𝐾)
30	25, 29	ressle 17428	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
31	28, 30	syl 18	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
32	31	adantl 486	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
33	26	adantl 486	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
34	33	sqxpeqd 5693	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐴 × 𝐴) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
35	32, 34	ineq12d 4174	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
36	35	dmeqd 5895	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = dom ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
37	24, 36, 33	3eqtr4d 2808	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = 𝐴)
38	13, 1	prsss 34306	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
39	38	dmeqd 5895	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ((le‘𝐾) ∩ (𝐴 × 𝐴)))
40	13, 1	prsdm 34304	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)
41	40	sseq2d 3969	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → (𝐴 ⊆ dom ≤ ↔ 𝐴 ⊆ 𝐵))
42	41	biimpar 482	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ dom ≤ )
43		sseqin2 4176	. . . . . . . 8 ⊢ (𝐴 ⊆ dom ≤ ↔ (dom ≤ ∩ 𝐴) = 𝐴)
44	42, 43	sylib 221	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (dom ≤ ∩ 𝐴) = 𝐴)
45	37, 39, 44	3eqtr4d 2808	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = (dom ≤ ∩ 𝐴))
46	4, 11	mp1i 14	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘ ≤ ) ∈ Top)
47	16	adantl 486	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V)
48		eqid 2763	. . . . . . . . . 10 ⊢ dom ≤ = dom ≤
49	48	ordttopon 23350	. . . . . . . . 9 ⊢ ( ≤ ∈ V → (ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ))
50	4, 49	mp1i 14	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ))
51		toponmax 23083	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ) → dom ≤ ∈ (ordTop‘ ≤ ))
52	50, 51	syl 18	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ≤ ∈ (ordTop‘ ≤ ))
53		elrestr 17476	. . . . . . 7 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ dom ≤ ∈ (ordTop‘ ≤ )) → (dom ≤ ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
54	46, 47, 52, 53	syl3anc 1398	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (dom ≤ ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
55	45, 54	eqeltrd 2863	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
56	55	snssd 4752	. . . 4 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → {dom ( ≤ ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
57		rabeq 3430	. . . . . . . . 9 ⊢ (dom ( ≤ ∩ (𝐴 × 𝐴)) = (dom ≤ ∩ 𝐴) → {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥})
58	45, 57	syl 18	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥})
59	45, 58	mpteq12dv 5198	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) = (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}))
60	59	rneqd 5928	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) = ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}))
61		inrab2 4270	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦 ≤ 𝑥}
62		inss2 4190	. . . . . . . . . . . . . 14 ⊢ (dom ≤ ∩ 𝐴) ⊆ 𝐴
63		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → 𝑦 ∈ (dom ≤ ∩ 𝐴))
64	62, 63	sselid 3935	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → 𝑦 ∈ 𝐴)
65		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → 𝑥 ∈ (dom ≤ ∩ 𝐴))
66	62, 65	sselid 3935	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → 𝑥 ∈ 𝐴)
67	66	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → 𝑥 ∈ 𝐴)
68		brinxp 5740	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 ≤ 𝑥 ↔ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥))
69	64, 67, 68	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (𝑦 ≤ 𝑥 ↔ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥))
70	69	notbid 321	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (¬ 𝑦 ≤ 𝑥 ↔ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥))
71	70	rabbidva 3422	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦 ≤ 𝑥} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥})
72	61, 71	eqtrid 2810	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥})
73	4, 11	mp1i 14	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → (ordTop‘ ≤ ) ∈ Top)
74	47	adantr 485	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → 𝐴 ∈ V)
75		simpl 487	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐾 ∈ Proset )
76		inss1 4189	. . . . . . . . . . . 12 ⊢ (dom ≤ ∩ 𝐴) ⊆ dom ≤
77	76	sseli 3933	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (dom ≤ ∩ 𝐴) → 𝑥 ∈ dom ≤ )
78	48	ordtopn1 23351	. . . . . . . . . . . . 13 ⊢ (( ≤ ∈ V ∧ 𝑥 ∈ dom ≤ ) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ ))
79	4, 78	mpan 702	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom ≤ → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ ))
80	79	adantl 486	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ dom ≤ ) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ ))
81	75, 77, 80	syl2an 607	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ ))
82		elrestr 17476	. . . . . . . . . 10 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ )) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
83	73, 74, 81, 82	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
84	72, 83	eqeltrrd 2864	. . . . . . . 8 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
85	84	fmpttd 7110	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}):(dom ≤ ∩ 𝐴)⟶((ordTop‘ ≤ ) ↾t 𝐴))
86	85	frnd 6714	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
87	60, 86	eqsstrd 3971	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
88		rabeq 3430	. . . . . . . . 9 ⊢ (dom ( ≤ ∩ (𝐴 × 𝐴)) = (dom ≤ ∩ 𝐴) → {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
89	45, 88	syl 18	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
90	45, 89	mpteq12dv 5198	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}) = (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}))
91	90	rneqd 5928	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}) = ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}))
92		inrab2 4270	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥 ≤ 𝑦}
93		brinxp 5740	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ≤ 𝑦 ↔ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦))
94	67, 64, 93	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (𝑥 ≤ 𝑦 ↔ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦))
95	94	notbid 321	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (¬ 𝑥 ≤ 𝑦 ↔ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦))
96	95	rabbidva 3422	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥 ≤ 𝑦} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
97	92, 96	eqtrid 2810	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
98	48	ordtopn2 23352	. . . . . . . . . . . . 13 ⊢ (( ≤ ∈ V ∧ 𝑥 ∈ dom ≤ ) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ ))
99	4, 98	mpan 702	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom ≤ → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ ))
100	99	adantl 486	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ dom ≤ ) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ ))
101	75, 77, 100	syl2an 607	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ ))
102		elrestr 17476	. . . . . . . . . 10 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ )) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
103	73, 74, 101, 102	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
104	97, 103	eqeltrrd 2864	. . . . . . . 8 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
105	104	fmpttd 7110	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}):(dom ≤ ∩ 𝐴)⟶((ordTop‘ ≤ ) ↾t 𝐴))
106	105	frnd 6714	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
107	91, 106	eqsstrd 3971	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
108	87, 107	unssd 4145	. . . 4 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
109	56, 108	unssd 4145	. . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ({dom ( ≤ ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
110		tgfiss 23148	. . 3 ⊢ ((((ordTop‘ ≤ ) ↾t 𝐴) ∈ Top ∧ ({dom ( ≤ ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴)) → (topGen‘(fi‘({dom ( ≤ ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}))))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
111	19, 109, 110	syl2anc 595	. 2 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (topGen‘(fi‘({dom ( ≤ ∩ (𝐴 × 𝐴))} ∪ (ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ∪ ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}))))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
112	10, 111	eqsstrd 3971	1 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  {csn 4589   class class class wbr 5109   ↦ cmpt 5192   × cxp 5659  dom cdm 5661  ran crn 5662  ‘cfv 6536  (class class class)co 7410  ficfi 9366  Basecbs 17264   ↾_s cress 17285  lecple 17312   ↾_t crest 17468  topGenctg 17485  ordTopcordt 17548   Proset cproset 18343  Topctop 23050  TopOnctopon 23067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9367  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-ple 17325  df-rest 17470  df-topgen 17491  df-ordt 17550  df-proset 18345  df-top 23051  df-topon 23068  df-bases 23103

This theorem is referenced by:  ordtrest2NEW  34313