Theorem ordtrestNEW 34255

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 6895	. . . . . 6 ⊢ (le‘𝐾) ∈ V
3	2	inex1 5288	. . . . 5 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
4	1, 3	eqeltri 2865	. . . 4 ⊢ ≤ ∈ V
5	4	inex1 5288	. . 3 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) ∈ V
6		eqid 2769	. . . 4 ⊢ dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ( ≤ ∩ (𝐴 × 𝐴))
7		eqid 2769	. . . 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 2769	. . . 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 23314	. . 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 23325	. . . . . 6 ⊢ ( ≤ ∈ V → (ordTop‘ ≤ ) ∈ Top)
12	4, 11	ax-mp 5	. . . . 5 ⊢ (ordTop‘ ≤ ) ∈ Top
13		ordtNEW.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
14		fvex 6895	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
15	13, 14	eqeltri 2865	. . . . . 6 ⊢ 𝐵 ∈ V
16	15	ssex 5292	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
17		resttop 23285	. . . . 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 33226	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Proset )
21		eqid 2769	. . . . . . . . . 10 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴))
22		eqid 2769	. . . . . . . . . 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 34248	. . . . . . . . 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 2769	. . . . . . . . . . . . . 14 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴)
26	25, 13	ressbas2 17297	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
27		fvex 6895	. . . . . . . . . . . . 13 ⊢ (Base‘(𝐾 ↾s 𝐴)) ∈ V
28	26, 27	eqeltrdi 2877	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
29		eqid 2769	. . . . . . . . . . . . 13 ⊢ (le‘𝐾) = (le‘𝐾)
30	25, 29	ressle 17432	. . . . . . . . . . . 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 5694	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐴 × 𝐴) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
35	32, 34	ineq12d 4182	. . . . . . . . 9 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
36	35	dmeqd 5896	. . . . . . . 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 2814	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = 𝐴)
38	13, 1	prsss 34250	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
39	38	dmeqd 5896	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ((le‘𝐾) ∩ (𝐴 × 𝐴)))
40	13, 1	prsdm 34248	. . . . . . . . . 10 ⊢ (𝐾 ∈ Proset → dom ≤ = 𝐵)
41	40	sseq2d 3977	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → (𝐴 ⊆ dom ≤ ↔ 𝐴 ⊆ 𝐵))
42	41	biimpar 482	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ dom ≤ )
43		sseqin2 4184	. . . . . . . 8 ⊢ (𝐴 ⊆ dom ≤ ↔ (dom ≤ ∩ 𝐴) = 𝐴)
44	42, 43	sylib 221	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (dom ≤ ∩ 𝐴) = 𝐴)
45	37, 39, 44	3eqtr4d 2814	. . . . . 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 2769	. . . . . . . . . 10 ⊢ dom ≤ = dom ≤
49	48	ordttopon 23318	. . . . . . . . 9 ⊢ ( ≤ ∈ V → (ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ))
50	4, 49	mp1i 14	. . . . . . . 8 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ))
51		toponmax 23051	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ∈ (TopOn‘dom ≤ ) → dom ≤ ∈ (ordTop‘ ≤ ))
52	50, 51	syl 18	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ≤ ∈ (ordTop‘ ≤ ))
53		elrestr 17480	. . . . . . 7 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ dom ≤ ∈ (ordTop‘ ≤ )) → (dom ≤ ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
54	46, 47, 52, 53	syl3anc 1396	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (dom ≤ ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
55	45, 54	eqeltrd 2869	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
56	55	snssd 4757	. . . 4 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → {dom ( ≤ ∩ (𝐴 × 𝐴))} ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
57		rabeq 3437	. . . . . . . . 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 5202	. . . . . . 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 5929	. . . . . 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 4278	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦 ≤ 𝑥}
62		inss2 4198	. . . . . . . . . . . . . 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 3943	. . . . . . . . . . . . 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 3943	. . . . . . . . . . . . . 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 5741	. . . . . . . . . . . . 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 3429	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦 ≤ 𝑥} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥})
72	61, 71	eqtrid 2816	. . . . . . . . 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 4197	. . . . . . . . . . . 12 ⊢ (dom ≤ ∩ 𝐴) ⊆ dom ≤
77	76	sseli 3941	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (dom ≤ ∩ 𝐴) → 𝑥 ∈ dom ≤ )
78	48	ordtopn1 23319	. . . . . . . . . . . . 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 17480	. . . . . . . . . 10 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∈ (ordTop‘ ≤ )) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
83	73, 74, 81, 82	syl3anc 1396	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑦 ≤ 𝑥} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
84	72, 83	eqeltrrd 2870	. . . . . . . 8 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥} ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
85	84	fmpttd 7111	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}):(dom ≤ ∩ 𝐴)⟶((ordTop‘ ≤ ) ↾t 𝐴))
86	85	frnd 6715	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
87	60, 86	eqsstrd 3979	. . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ↦ {𝑦 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑦( ≤ ∩ (𝐴 × 𝐴))𝑥}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
88		rabeq 3437	. . . . . . . . 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 5202	. . . . . . 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 5929	. . . . . 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 4278	. . . . . . . . . 10 ⊢ ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥 ≤ 𝑦}
93		brinxp 5741	. . . . . . . . . . . . 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 3429	. . . . . . . . . 10 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥 ≤ 𝑦} = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
97	92, 96	eqtrid 2816	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) = {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦})
98	48	ordtopn2 23320	. . . . . . . . . . . . 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 17480	. . . . . . . . . 10 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ 𝐴 ∈ V ∧ {𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∈ (ordTop‘ ≤ )) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
103	73, 74, 101, 102	syl3anc 1396	. . . . . . . . 9 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → ({𝑦 ∈ dom ≤ ∣ ¬ 𝑥 ≤ 𝑦} ∩ 𝐴) ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
104	97, 103	eqeltrrd 2870	. . . . . . . 8 ⊢ (((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) ∧ 𝑥 ∈ (dom ≤ ∩ 𝐴)) → {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦} ∈ ((ordTop‘ ≤ ) ↾t 𝐴))
105	104	fmpttd 7111	. . . . . . 7 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}):(dom ≤ ∩ 𝐴)⟶((ordTop‘ ≤ ) ↾t 𝐴))
106	105	frnd 6715	. . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ran (𝑥 ∈ (dom ≤ ∩ 𝐴) ↦ {𝑦 ∈ (dom ≤ ∩ 𝐴) ∣ ¬ 𝑥( ≤ ∩ (𝐴 × 𝐴))𝑦}) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
107	91, 106	eqsstrd 3979	. . . . 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 4153	. . . 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 4153	. . 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 23116	. . 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 3979	1 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  {csn 4594   class class class wbr 5113   ↦ cmpt 5196   × cxp 5660  dom cdm 5662  ran crn 5663  ‘cfv 6537  (class class class)co 7411  ficfi 9369  Basecbs 17268   ↾_s cress 17289  lecple 17316   ↾_t crest 17472  topGenctg 17489  ordTopcordt 17552   Proset cproset 18347  Topctop 23018  TopOnctopon 23035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fi 9370  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-3 12303  df-4 12304  df-5 12305  df-6 12306  df-7 12307  df-8 12308  df-9 12309  df-n0 12504  df-z 12591  df-dec 12711  df-sets 17223  df-slot 17241  df-ndx 17253  df-base 17269  df-ress 17290  df-ple 17329  df-rest 17474  df-topgen 17495  df-ordt 17554  df-proset 18349  df-top 23019  df-topon 23036  df-bases 23071

This theorem is referenced by:  ordtrest2NEW  34257