Theorem ordtrestNEW 31773

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 6769	. . . . . 6 ⊢ (le‘𝐾) ∈ V
3	2	inex1 5236	. . . . 5 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
4	1, 3	eqeltri 2835	. . . 4 ⊢ ≤ ∈ V
5	4	inex1 5236	. . 3 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) ∈ V
6		eqid 2738	. . . 4 ⊢ dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ( ≤ ∩ (𝐴 × 𝐴))
7		eqid 2738	. . . 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 2738	. . . 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 22248	. . 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 13	. 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 22259	. . . . . 6 ⊢ ( ≤ ∈ V → (ordTop‘ ≤ ) ∈ Top)
12	4, 11	ax-mp 5	. . . . 5 ⊢ (ordTop‘ ≤ ) ∈ Top
13		ordtNEW.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
14		fvex 6769	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
15	13, 14	eqeltri 2835	. . . . . 6 ⊢ 𝐵 ∈ V
16	15	ssex 5240	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
17		resttop 22219	. . . . 5 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V) → ((ordTop‘ ≤ ) ↾t 𝐴) ∈ Top)
18	12, 16, 17	sylancr 586	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((ordTop‘ ≤ ) ↾t 𝐴) ∈ Top)
19	18	adantl 481	. . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((ordTop‘ ≤ ) ↾t 𝐴) ∈ Top)
20	13	ressprs 31143	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Proset )
21		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴))
22		eqid 2738	. . . . . . . . . 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 31766	. . . . . . . . 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 17	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) = (Base‘(𝐾 ↾s 𝐴)))
25		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴)
26	25, 13	ressbas2 16875	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
27		fvex 6769	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐾 ↾s 𝐴)) ∈ V
28	26, 27	eqeltrdi 2847	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
29		eqid 2738	. . . . . . . . . . . . 13 ⊢ (le‘𝐾) = (le‘𝐾)
30	25, 29	ressle 17013	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
31	28, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝐵 → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
32	31	adantl 481	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
33	26	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
34	33	sqxpeqd 5612	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐴 × 𝐴) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
35	32, 34	ineq12d 4144	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
36	35	dmeqd 5803	. . . . . . . 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 2788	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = 𝐴)
38	13, 1	prsss 31768	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
39	38	dmeqd 5803	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ((le‘𝐾) ∩ (𝐴 × 𝐴)))
40	13, 1	prsdm 31766	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)
41	40	sseq2d 3949	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → (𝐴 ⊆ dom ≤ ↔ 𝐴 ⊆ 𝐵))
42	41	biimpar 477	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ dom ≤ )
43		sseqin2 4146	. . . . . . . 8 ⊢ (𝐴 ⊆ dom ≤ ↔ (dom ≤ ∩ 𝐴) = 𝐴)
44	42, 43	sylib 217	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (dom ≤ ∩ 𝐴) = 𝐴)
45	37, 39, 44	3eqtr4d 2788	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = (dom ≤ ∩ 𝐴))
46	4, 11	mp1i 13	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘ ≤ ) ∈ Top)
47	16	adantl 481	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ V)
48		eqid 2738	. . . . . . . . . 10 ⊢ dom ≤ = dom ≤
49	48	ordttopon 22252	. . . . . . . . 9 ⊢ ( ≤ ∈ V → (ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ))
50	4, 49	mp1i 13	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ))
51		toponmax 21983	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ) → dom ≤ ∈ (ordTop‘ ≤ ))
52	50, 51	syl 17	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ≤ ∈ (ordTop‘ ≤ ))
53		elrestr 17056	. . . . . . 7 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ dom ≤ ∈ (ordTop‘ ≤ )) → (dom ≤ ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
54	46, 47, 52, 53	syl3anc 1369	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (dom ≤ ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
55	45, 54	eqeltrd 2839	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
56	55	snssd 4739	. . . 4 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → {dom ( ≤ ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
57		rabeq 3408	. . . . . . . . 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 17	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥})
59	45, 58	mpteq12dv 5161	. . . . . . 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 5836	. . . . . 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 4238	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦 ≤ 𝑥}
62		inss2 4160	. . . . . . . . . . . . . 14 ⊢ (dom ≤ ∩ 𝐴) ⊆ 𝐴
63		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → 𝑦 ∈ (dom ≤ ∩ 𝐴))
64	62, 63	sselid 3915	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → 𝑦 ∈ 𝐴)
65		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → 𝑥 ∈ (dom ≤ ∩ 𝐴))
66	62, 65	sselid 3915	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → 𝑥 ∈ 𝐴)
67	66	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → 𝑥 ∈ 𝐴)
68		brinxp 5656	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 ≤ 𝑥 ↔ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥))
69	64, 67, 68	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (𝑦 ≤ 𝑥 ↔ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥))
70	69	notbid 317	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (¬ 𝑦 ≤ 𝑥 ↔ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥))
71	70	rabbidva 3402	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦 ≤ 𝑥} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥})
72	61, 71	syl5eq 2791	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥})
73	4, 11	mp1i 13	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → (ordTop‘ ≤ ) ∈ Top)
74	47	adantr 480	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → 𝐴 ∈ V)
75		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐾 ∈ Proset )
76		inss1 4159	. . . . . . . . . . . 12 ⊢ (dom ≤ ∩ 𝐴) ⊆ dom ≤
77	76	sseli 3913	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (dom ≤ ∩ 𝐴) → 𝑥 ∈ dom ≤ )
78	48	ordtopn1 22253	. . . . . . . . . . . . 13 ⊢ (( ≤ ∈ V ∧ 𝑥 ∈ dom ≤ ) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ ))
79	4, 78	mpan 686	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom ≤ → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ ))
80	79	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ dom ≤ ) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ ))
81	75, 77, 80	syl2an 595	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ ))
82		elrestr 17056	. . . . . . . . . 10 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ )) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
83	73, 74, 81, 82	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
84	72, 83	eqeltrrd 2840	. . . . . . . 8 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
85	84	fmpttd 6971	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}):(dom ≤ ∩ 𝐴)⟶((ordTop‘ ≤ ) ↾t 𝐴))
86	85	frnd 6592	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
87	60, 86	eqsstrd 3955	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
88		rabeq 3408	. . . . . . . . 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 17	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
90	45, 89	mpteq12dv 5161	. . . . . . 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 5836	. . . . . 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 4238	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥 ≤ 𝑦}
93		brinxp 5656	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ≤ 𝑦 ↔ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦))
94	67, 64, 93	syl2anc 583	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (𝑥 ≤ 𝑦 ↔ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦))
95	94	notbid 317	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (¬ 𝑥 ≤ 𝑦 ↔ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦))
96	95	rabbidva 3402	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥 ≤ 𝑦} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
97	92, 96	syl5eq 2791	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
98	48	ordtopn2 22254	. . . . . . . . . . . . 13 ⊢ (( ≤ ∈ V ∧ 𝑥 ∈ dom ≤ ) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ ))
99	4, 98	mpan 686	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom ≤ → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ ))
100	99	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝑥 ∈ dom ≤ ) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ ))
101	75, 77, 100	syl2an 595	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ ))
102		elrestr 17056	. . . . . . . . . 10 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ )) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
103	73, 74, 101, 102	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
104	97, 103	eqeltrrd 2840	. . . . . . . 8 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
105	104	fmpttd 6971	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}):(dom ≤ ∩ 𝐴)⟶((ordTop‘ ≤ ) ↾t 𝐴))
106	105	frnd 6592	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
107	91, 106	eqsstrd 3955	. . . . 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 4116	. . . 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 4116	. . 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 22049	. . 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 583	. 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 3955	1 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  {csn 4558   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  dom cdm 5580  ran crn 5581  ‘cfv 6418  (class class class)co 7255  ficfi 9099  Basecbs 16840   ↾_s cress 16867  lecple 16895   ↾_t crest 17048  topGenctg 17065  ordTopcordt 17127   Proset cproset 17926  Topctop 21950  TopOnctopon 21967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fi 9100  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-dec 12367  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-ple 16908  df-rest 17050  df-topgen 17071  df-ordt 17129  df-proset 17928  df-top 21951  df-topon 21968  df-bases 22004

This theorem is referenced by:  ordtrest2NEW  31775