Theorem ordtrest2NEW 29297

Description: An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in ℝ, but in other sets like ℚ there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (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))
ordtrest2NEW.2	⊢ (𝜑 → 𝐾 ∈ Toset)
ordtrest2NEW.3	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
ordtrest2NEW.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴)

Assertion

Ref	Expression
ordtrest2NEW	⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾t 𝐴))

Distinct variable groups:   𝑥,𝑦, ≤   𝑥,𝐵,𝑦   𝑥,𝐾,𝑦   𝑥,𝐴,𝑦,𝑧   𝑧, ≤   𝑧,𝐴   𝑧,𝐵   𝜑,𝑥,𝑦,𝑧   𝑧,𝐾

Proof of Theorem ordtrest2NEW

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtrest2NEW.2	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Toset)
2		tospos 28989	. . . 4 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset)
3		posprs 16772	. . . 4 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Preset )
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝜑 → 𝐾 ∈ Preset )
5		ordtrest2NEW.3	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
6		ordtNEW.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
7		ordtNEW.l	. . . 4 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
8	6, 7	ordtrestNEW 29295	. . 3 ⊢ ((𝐾 ∈ Preset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
9	4, 5, 8	syl2anc 691	. 2 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
10		eqid 2610	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) = ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})
11		eqid 2610	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}) = ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})
12	6, 7, 10, 11	ordtprsval 29292	. . . . . . 7 ⊢ (𝐾 ∈ Preset → (ordTop‘ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))))
13	4, 12	syl 17	. . . . . 6 ⊢ (𝜑 → (ordTop‘ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))))
14	13	oveq1d 6564	. . . . 5 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾t 𝐴) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))) ↾t 𝐴))
15		fibas 20592	. . . . . 6 ⊢ (fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))) ∈ TopBases
16		fvex 6113	. . . . . . . . 9 ⊢ (Base‘𝐾) ∈ V
17	6, 16	eqeltri 2684	. . . . . . . 8 ⊢ 𝐵 ∈ V
18	17	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
19	18, 5	ssexd 4733	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
20		tgrest 20773	. . . . . 6 ⊢ (((fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))) ∈ TopBases ∧ 𝐴 ∈ V) → (topGen‘((fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))) ↾t 𝐴))
21	15, 19, 20	sylancr 694	. . . . 5 ⊢ (𝜑 → (topGen‘((fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))) ↾t 𝐴))
22	14, 21	eqtr4d 2647	. . . 4 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾t 𝐴) = (topGen‘((fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))) ↾t 𝐴)))
23		firest 15916	. . . . 5 ⊢ (fi‘(({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴)) = ((fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))) ↾t 𝐴)
24	23	fveq2i 6106	. . . 4 ⊢ (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴))) = (topGen‘((fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))) ↾t 𝐴))
25	22, 24	syl6eqr 2662	. . 3 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾t 𝐴) = (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴))))
26		fvex 6113	. . . . . . . 8 ⊢ (le‘𝐾) ∈ V
27	26	inex1 4727	. . . . . . 7 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
28	7, 27	eqeltri 2684	. . . . . 6 ⊢ ≤ ∈ V
29	28	inex1 4727	. . . . 5 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) ∈ V
30		ordttop 20814	. . . . 5 ⊢ (( ≤ ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
31	29, 30	mp1i 13	. . . 4 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
32	6, 7, 10, 11	ordtprsuni 29293	. . . . . . . . 9 ⊢ (𝐾 ∈ Preset → 𝐵 = ∪ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))
33	4, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = ∪ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))
34	33, 18	eqeltrrd 2689	. . . . . . 7 ⊢ (𝜑 → ∪ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ∈ V)
35		uniexb 6866	. . . . . . 7 ⊢ (({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ∈ V ↔ ∪ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ∈ V)
36	34, 35	sylibr 223	. . . . . 6 ⊢ (𝜑 → ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ∈ V)
37		restval 15910	. . . . . 6 ⊢ ((({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ∈ V ∧ 𝐴 ∈ V) → (({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↦ (𝑣 ∩ 𝐴)))
38	36, 19, 37	syl2anc 691	. . . . 5 ⊢ (𝜑 → (({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↦ (𝑣 ∩ 𝐴)))
39		sseqin2 3779	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
40	5, 39	sylib 207	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴)
41		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ( ≤ ∩ (𝐴 × 𝐴))
42	41	ordttopon 20807	. . . . . . . . . . . . . 14 ⊢ (( ≤ ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))))
43	29, 42	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))))
44	6, 7	prsssdm 29291	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Preset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
45	4, 5, 44	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
46	45	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝜑 → (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
47	43, 46	eleqtrd 2690	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
48		toponmax 20543	. . . . . . . . . . . 12 ⊢ ((ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
49	47, 48	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
50	40, 49	eqeltrd 2688	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
51		elsni 4142	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ {𝐵} → 𝑣 = 𝐵)
52	51	ineq1d 3775	. . . . . . . . . . 11 ⊢ (𝑣 ∈ {𝐵} → (𝑣 ∩ 𝐴) = (𝐵 ∩ 𝐴))
53	52	eleq1d 2672	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝐵} → ((𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ (𝐵 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
54	50, 53	syl5ibrcom 236	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ {𝐵} → (𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
55	54	ralrimiv 2948	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ {𝐵} (𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
56		ordtrest2NEW.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴)
57	6, 7, 1, 5, 56	ordtrest2NEWlem 29296	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
58		eqid 2610	. . . . . . . . . . . 12 ⊢ (ODual‘𝐾) = (ODual‘𝐾)
59	58, 6	odubas 16956	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘(ODual‘𝐾))
60	7	cnveqi 5219	. . . . . . . . . . . 12 ⊢ ◡ ≤ = ◡((le‘𝐾) ∩ (𝐵 × 𝐵))
61		cnvin 5459	. . . . . . . . . . . . 13 ⊢ ◡((le‘𝐾) ∩ (𝐵 × 𝐵)) = (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵))
62		cnvxp 5470	. . . . . . . . . . . . . 14 ⊢ ◡(𝐵 × 𝐵) = (𝐵 × 𝐵)
63	62	ineq2i 3773	. . . . . . . . . . . . 13 ⊢ (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵)) = (◡(le‘𝐾) ∩ (𝐵 × 𝐵))
64		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (le‘𝐾) = (le‘𝐾)
65	58, 64	oduleval 16954	. . . . . . . . . . . . . 14 ⊢ ◡(le‘𝐾) = (le‘(ODual‘𝐾))
66	65	ineq1i 3772	. . . . . . . . . . . . 13 ⊢ (◡(le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
67	61, 63, 66	3eqtri 2636	. . . . . . . . . . . 12 ⊢ ◡((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
68	60, 67	eqtri 2632	. . . . . . . . . . 11 ⊢ ◡ ≤ = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
69	58	odutos 28994	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Toset → (ODual‘𝐾) ∈ Toset)
70	1, 69	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ODual‘𝐾) ∈ Toset)
71		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
72		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
73	71, 72	brcnv 5227	. . . . . . . . . . . . . . . 16 ⊢ (𝑦◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦)
74		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
75	72, 74	brcnv 5227	. . . . . . . . . . . . . . . 16 ⊢ (𝑧◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧)
76	73, 75	anbi12ci 730	. . . . . . . . . . . . . . 15 ⊢ ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))
77	76	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐵 → ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)))
78	77	rabbiia 3161	. . . . . . . . . . . . 13 ⊢ {𝑧 ∈ 𝐵 ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} = {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}
79	78, 56	syl5eqss 3612	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴)
80	79	ancom2s 840	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴)
81	59, 68, 70, 5, 80	ordtrest2NEWlem 29296	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))))
82		vex 3176	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑤 ∈ V
83	82, 72	brcnv 5227	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑤)
84	83	bicomi 213	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ≤ 𝑤 ↔ 𝑤◡ ≤ 𝑧)
85	84	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ 𝑤◡ ≤ 𝑧))
86	85	notbid 307	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ 𝑧 ≤ 𝑤 ↔ ¬ 𝑤◡ ≤ 𝑧))
87	86	rabbidv 3164	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤} = {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧})
88	87	mpteq2dv 4673	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}) = (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧}))
89	88	rneqd 5274	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}) = ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧}))
90		cnvin 5459	. . . . . . . . . . . . . . 15 ⊢ ◡( ≤ ∩ (𝐴 × 𝐴)) = (◡ ≤ ∩ ◡(𝐴 × 𝐴))
91		cnvxp 5470	. . . . . . . . . . . . . . . 16 ⊢ ◡(𝐴 × 𝐴) = (𝐴 × 𝐴)
92	91	ineq2i 3773	. . . . . . . . . . . . . . 15 ⊢ (◡ ≤ ∩ ◡(𝐴 × 𝐴)) = (◡ ≤ ∩ (𝐴 × 𝐴))
93	90, 92	eqtri 2632	. . . . . . . . . . . . . 14 ⊢ ◡( ≤ ∩ (𝐴 × 𝐴)) = (◡ ≤ ∩ (𝐴 × 𝐴))
94	93	fveq2i 6106	. . . . . . . . . . . . 13 ⊢ (ordTop‘◡( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴)))
95	6	ressprs 28986	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Preset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Preset )
96	4, 5, 95	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾 ↾s 𝐴) ∈ Preset )
97		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴))
98		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
99	97, 98	ordtcnvNEW 29294	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ↾s 𝐴) ∈ Preset → (ordTop‘◡((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))) = (ordTop‘((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))))
100	96, 99	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ordTop‘◡((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))) = (ordTop‘((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))))
101	6, 7	prsss 29290	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Preset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
102	4, 5, 101	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
103		eqid 2610	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴)
104	103, 64	ressle 15882	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
105	19, 104	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
106	103, 6	ressbas2 15758	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
107	5, 106	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
108	107	sqxpeqd 5065	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 × 𝐴) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
109	105, 108	ineq12d 3777	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
110	102, 109	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
111	110	cnveqd 5220	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ◡( ≤ ∩ (𝐴 × 𝐴)) = ◡((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
112	111	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ordTop‘◡( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘◡((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))))
113	110	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))))
114	100, 112, 113	3eqtr4d 2654	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘◡( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
115	94, 114	syl5reqr 2659	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))))
116	115	eleq2d 2673	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∩ 𝐴) ∈ (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴)))))
117	89, 116	raleqbidv 3129	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴)))))
118	81, 117	mpbird 246	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
119		ralunb 3756	. . . . . . . . 9 ⊢ (∀𝑣 ∈ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
120	57, 118, 119	sylanbrc 695	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
121		ralunb 3756	. . . . . . . 8 ⊢ (∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ {𝐵} (𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
122	55, 120, 121	sylanbrc 695	. . . . . . 7 ⊢ (𝜑 → ∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
123		eqid 2610	. . . . . . . 8 ⊢ (𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↦ (𝑣 ∩ 𝐴)) = (𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↦ (𝑣 ∩ 𝐴))
124	123	fmpt 6289	. . . . . . 7 ⊢ (∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↦ (𝑣 ∩ 𝐴)):({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))⟶(ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
125	122, 124	sylib 207	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↦ (𝑣 ∩ 𝐴)):({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))⟶(ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
126		frn 5966	. . . . . 6 ⊢ ((𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↦ (𝑣 ∩ 𝐴)):({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))⟶(ordTop‘( ≤ ∩ (𝐴 × 𝐴))) → ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↦ (𝑣 ∩ 𝐴)) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
127	125, 126	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↦ (𝑣 ∩ 𝐴)) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
128	38, 127	eqsstrd 3602	. . . 4 ⊢ (𝜑 → (({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
129		tgfiss 20606	. . . 4 ⊢ (((ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top ∧ (({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))) → (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
130	31, 128, 129	syl2anc 691	. . 3 ⊢ (𝜑 → (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
131	25, 130	eqsstrd 3602	. 2 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾t 𝐴) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
132	9, 131	eqssd 3585	1 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾t 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ficfi 8199  Basecbs 15695   ↾_s cress 15696  lecple 15775   ↾_t crest 15904  topGenctg 15921  ordTopcordt 15982   Preset cpreset 16749  Posetcpo 16763  Tosetctos 16856  ODualcodu 16951  Topctop 20517  TopOnctopon 20518  TopBasesctb 20520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-ple 15788  df-rest 15906  df-topgen 15927  df-ordt 15984  df-preset 16751  df-poset 16769  df-toset 16857  df-odu 16952  df-top 20521  df-bases 20522  df-topon 20523

This theorem is referenced by: (None)