Theorem ordtrestNEW 34065

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 6853	. . . . . 6 ⊢ (le‘𝐾) ∈ V
3	2	inex1 5258	. . . . 5 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
4	1, 3	eqeltri 2832	. . . 4 ⊢ ≤ ∈ V
5	4	inex1 5258	. . 3 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) ∈ V
6		eqid 2736	. . . 4 ⊢ dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ( ≤ ∩ (𝐴 × 𝐴))
7		eqid 2736	. . . 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 2736	. . . 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 23154	. . 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 23165	. . . . . 6 ⊢ ( ≤ ∈ V → (ordTop‘ ≤ ) ∈ Top)
12	4, 11	ax-mp 5	. . . . 5 ⊢ (ordTop‘ ≤ ) ∈ Top
13		ordtNEW.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
14		fvex 6853	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
15	13, 14	eqeltri 2832	. . . . . 6 ⊢ 𝐵 ∈ V
16	15	ssex 5262	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
17		resttop 23125	. . . . 5 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V) → ((ordTop‘ ≤ ) ↾t 𝐴) ∈ Top)
18	12, 16, 17	sylancr 588	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((ordTop‘ ≤ ) ↾t 𝐴) ∈ Top)
19	18	adantl 481	. . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((ordTop‘ ≤ ) ↾t 𝐴) ∈ Top)
20	13	ressprs 33026	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Proset )
21		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴))
22		eqid 2736	. . . . . . . . . 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 34058	. . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴)
26	25, 13	ressbas2 17208	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
27		fvex 6853	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐾 ↾s 𝐴)) ∈ V
28	26, 27	eqeltrdi 2844	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
29		eqid 2736	. . . . . . . . . . . . 13 ⊢ (le‘𝐾) = (le‘𝐾)
30	25, 29	ressle 17343	. . . . . . . . . . . 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 5663	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐴 × 𝐴) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
35	32, 34	ineq12d 4161	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
36	35	dmeqd 5860	. . . . . . . 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 2781	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = 𝐴)
38	13, 1	prsss 34060	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
39	38	dmeqd 5860	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ((le‘𝐾) ∩ (𝐴 × 𝐴)))
40	13, 1	prsdm 34058	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)
41	40	sseq2d 3954	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → (𝐴 ⊆ dom ≤ ↔ 𝐴 ⊆ 𝐵))
42	41	biimpar 477	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ dom ≤ )
43		sseqin2 4163	. . . . . . . 8 ⊢ (𝐴 ⊆ dom ≤ ↔ (dom ≤ ∩ 𝐴) = 𝐴)
44	42, 43	sylib 218	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (dom ≤ ∩ 𝐴) = 𝐴)
45	37, 39, 44	3eqtr4d 2781	. . . . . 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 2736	. . . . . . . . . 10 ⊢ dom ≤ = dom ≤
49	48	ordttopon 23158	. . . . . . . . 9 ⊢ ( ≤ ∈ V → (ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ))
50	4, 49	mp1i 13	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ))
51		toponmax 22891	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ) → dom ≤ ∈ (ordTop‘ ≤ ))
52	50, 51	syl 17	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ≤ ∈ (ordTop‘ ≤ ))
53		elrestr 17391	. . . . . . 7 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ dom ≤ ∈ (ordTop‘ ≤ )) → (dom ≤ ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
54	46, 47, 52, 53	syl3anc 1374	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (dom ≤ ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
55	45, 54	eqeltrd 2836	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
56	55	snssd 4730	. . . 4 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → {dom ( ≤ ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
57		rabeq 3403	. . . . . . . . 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 5172	. . . . . . 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 5893	. . . . . 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 4257	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦 ≤ 𝑥}
62		inss2 4178	. . . . . . . . . . . . . 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 3919	. . . . . . . . . . . . 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 3919	. . . . . . . . . . . . . 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 5710	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦 ≤ 𝑥 ↔ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥))
69	64, 67, 68	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (𝑦 ≤ 𝑥 ↔ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥))
70	69	notbid 318	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (¬ 𝑦 ≤ 𝑥 ↔ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥))
71	70	rabbidva 3395	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦 ≤ 𝑥} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥})
72	61, 71	eqtrid 2783	. . . . . . . . 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 4177	. . . . . . . . . . . 12 ⊢ (dom ≤ ∩ 𝐴) ⊆ dom ≤
77	76	sseli 3917	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (dom ≤ ∩ 𝐴) → 𝑥 ∈ dom ≤ )
78	48	ordtopn1 23159	. . . . . . . . . . . . 13 ⊢ (( ≤ ∈ V ∧ 𝑥 ∈ dom ≤ ) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ ))
79	4, 78	mpan 691	. . . . . . . . . . . 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 597	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ ))
82		elrestr 17391	. . . . . . . . . 10 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ )) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
83	73, 74, 81, 82	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
84	72, 83	eqeltrrd 2837	. . . . . . . 8 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
85	84	fmpttd 7067	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}):(dom ≤ ∩ 𝐴)⟶((ordTop‘ ≤ ) ↾t 𝐴))
86	85	frnd 6676	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
87	60, 86	eqsstrd 3956	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
88		rabeq 3403	. . . . . . . . 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 5172	. . . . . . 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 5893	. . . . . 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 4257	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥 ≤ 𝑦}
93		brinxp 5710	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ≤ 𝑦 ↔ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦))
94	67, 64, 93	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (𝑥 ≤ 𝑦 ↔ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦))
95	94	notbid 318	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) ∧ 𝑦 ∈ (dom ≤ ∩ 𝐴)) → (¬ 𝑥 ≤ 𝑦 ↔ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦))
96	95	rabbidva 3395	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥 ≤ 𝑦} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
97	92, 96	eqtrid 2783	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
98	48	ordtopn2 23160	. . . . . . . . . . . . 13 ⊢ (( ≤ ∈ V ∧ 𝑥 ∈ dom ≤ ) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ ))
99	4, 98	mpan 691	. . . . . . . . . . . 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 597	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ ))
102		elrestr 17391	. . . . . . . . . 10 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ )) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
103	73, 74, 101, 102	syl3anc 1374	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
104	97, 103	eqeltrrd 2837	. . . . . . . 8 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
105	104	fmpttd 7067	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}):(dom ≤ ∩ 𝐴)⟶((ordTop‘ ≤ ) ↾t 𝐴))
106	105	frnd 6676	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
107	91, 106	eqsstrd 3956	. . . . 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 4132	. . . 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 4132	. . 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 22956	. . 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 585	. 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 3956	1 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3389  Vcvv 3429   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  {csn 4567   class class class wbr 5085   ↦ cmpt 5166   × cxp 5629  dom cdm 5631  ran crn 5632  ‘cfv 6498  (class class class)co 7367  ficfi 9323  Basecbs 17179   ↾_s cress 17200  lecple 17227   ↾_t crest 17383  topGenctg 17400  ordTopcordt 17463   Proset cproset 18258  Topctop 22858  TopOnctopon 22875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fi 9324  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-dec 12645  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-ple 17240  df-rest 17385  df-topgen 17406  df-ordt 17465  df-proset 18260  df-top 22859  df-topon 22876  df-bases 22911

This theorem is referenced by:  ordtrest2NEW  34067