Theorem ordtrestNEW 34099

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 6855	. . . . . 6 ⊢ (le‘𝐾) ∈ V
3	2	inex1 5264	. . . . 5 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
4	1, 3	eqeltri 2833	. . . 4 ⊢ ≤ ∈ V
5	4	inex1 5264	. . 3 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) ∈ V
6		eqid 2737	. . . 4 ⊢ dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ( ≤ ∩ (𝐴 × 𝐴))
7		eqid 2737	. . . 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 2737	. . . 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 23145	. . 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 23156	. . . . . 6 ⊢ ( ≤ ∈ V → (ordTop‘ ≤ ) ∈ Top)
12	4, 11	ax-mp 5	. . . . 5 ⊢ (ordTop‘ ≤ ) ∈ Top
13		ordtNEW.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
14		fvex 6855	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
15	13, 14	eqeltri 2833	. . . . . 6 ⊢ 𝐵 ∈ V
16	15	ssex 5268	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
17		resttop 23116	. . . . 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 33059	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Proset )
21		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴))
22		eqid 2737	. . . . . . . . . 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 34092	. . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴)
26	25, 13	ressbas2 17177	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
27		fvex 6855	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐾 ↾s 𝐴)) ∈ V
28	26, 27	eqeltrdi 2845	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
29		eqid 2737	. . . . . . . . . . . . 13 ⊢ (le‘𝐾) = (le‘𝐾)
30	25, 29	ressle 17312	. . . . . . . . . . . 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 5664	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐴 × 𝐴) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
35	32, 34	ineq12d 4175	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
36	35	dmeqd 5862	. . . . . . . 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 2782	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = 𝐴)
38	13, 1	prsss 34094	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
39	38	dmeqd 5862	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ((le‘𝐾) ∩ (𝐴 × 𝐴)))
40	13, 1	prsdm 34092	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)
41	40	sseq2d 3968	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → (𝐴 ⊆ dom ≤ ↔ 𝐴 ⊆ 𝐵))
42	41	biimpar 477	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ dom ≤ )
43		sseqin2 4177	. . . . . . . 8 ⊢ (𝐴 ⊆ dom ≤ ↔ (dom ≤ ∩ 𝐴) = 𝐴)
44	42, 43	sylib 218	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (dom ≤ ∩ 𝐴) = 𝐴)
45	37, 39, 44	3eqtr4d 2782	. . . . . 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 2737	. . . . . . . . . 10 ⊢ dom ≤ = dom ≤
49	48	ordttopon 23149	. . . . . . . . 9 ⊢ ( ≤ ∈ V → (ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ))
50	4, 49	mp1i 13	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ))
51		toponmax 22882	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ) → dom ≤ ∈ (ordTop‘ ≤ ))
52	50, 51	syl 17	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ≤ ∈ (ordTop‘ ≤ ))
53		elrestr 17360	. . . . . . 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 2837	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
56	55	snssd 4767	. . . 4 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → {dom ( ≤ ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
57		rabeq 3415	. . . . . . . . 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 5187	. . . . . . 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 5895	. . . . . 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 4271	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦 ≤ 𝑥}
62		inss2 4192	. . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . 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 5711	. . . . . . . . . . . . 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 3407	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦 ≤ 𝑥} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥})
72	61, 71	eqtrid 2784	. . . . . . . . 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 4191	. . . . . . . . . . . 12 ⊢ (dom ≤ ∩ 𝐴) ⊆ dom ≤
77	76	sseli 3931	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (dom ≤ ∩ 𝐴) → 𝑥 ∈ dom ≤ )
78	48	ordtopn1 23150	. . . . . . . . . . . . 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 17360	. . . . . . . . . 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 2838	. . . . . . . 8 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
85	84	fmpttd 7069	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}):(dom ≤ ∩ 𝐴)⟶((ordTop‘ ≤ ) ↾t 𝐴))
86	85	frnd 6678	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
87	60, 86	eqsstrd 3970	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
88		rabeq 3415	. . . . . . . . 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 5187	. . . . . . 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 5895	. . . . . 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 4271	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥 ≤ 𝑦}
93		brinxp 5711	. . . . . . . . . . . . 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 3407	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥 ≤ 𝑦} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
97	92, 96	eqtrid 2784	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
98	48	ordtopn2 23151	. . . . . . . . . . . . 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 17360	. . . . . . . . . 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 2838	. . . . . . . 8 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
105	104	fmpttd 7069	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}):(dom ≤ ∩ 𝐴)⟶((ordTop‘ ≤ ) ↾t 𝐴))
106	105	frnd 6678	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
107	91, 106	eqsstrd 3970	. . . . 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 4146	. . . 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 4146	. . 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 22947	. . 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 3970	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 3401  Vcvv 3442   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  {csn 4582   class class class wbr 5100   ↦ cmpt 5181   × cxp 5630  dom cdm 5632  ran crn 5633  ‘cfv 6500  (class class class)co 7368  ficfi 9325  Basecbs 17148   ↾_s cress 17169  lecple 17196   ↾_t crest 17352  topGenctg 17369  ordTopcordt 17432   Proset cproset 18227  Topctop 22849  TopOnctopon 22866

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fi 9326  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-dec 12620  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-ple 17209  df-rest 17354  df-topgen 17375  df-ordt 17434  df-proset 18229  df-top 22850  df-topon 22867  df-bases 22902

This theorem is referenced by:  ordtrest2NEW  34101