Theorem ordtrest2NEW 33869

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 18490	. . . 4 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset)
3		posprs 18386	. . . 4 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset )
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝜑 → 𝐾 ∈ Proset )
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 33867	. . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
9	4, 5, 8	syl2anc 583	. 2 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
10		eqid 2740	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) = ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})
11		eqid 2740	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}) = ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})
12	6, 7, 10, 11	ordtprsval 33864	. . . . . . 7 ⊢ (𝐾 ∈ Proset → (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 7463	. . . . 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 23005	. . . . . 6 ⊢ (fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))) ∈ TopBases
16	6	fvexi 6934	. . . . . . . 8 ⊢ 𝐵 ∈ V
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
18	17, 5	ssexd 5342	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
19		tgrest 23188	. . . . . 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))
20	15, 18, 19	sylancr 586	. . . . 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))
21	14, 20	eqtr4d 2783	. . . 4 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾t 𝐴) = (topGen‘((fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))) ↾t 𝐴)))
22		firest 17492	. . . . 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)
23	22	fveq2i 6923	. . . 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))
24	21, 23	eqtr4di 2798	. . 3 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾t 𝐴) = (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴))))
25		fvex 6933	. . . . . . . 8 ⊢ (le‘𝐾) ∈ V
26	25	inex1 5335	. . . . . . 7 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
27	7, 26	eqeltri 2840	. . . . . 6 ⊢ ≤ ∈ V
28	27	inex1 5335	. . . . 5 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) ∈ V
29		ordttop 23229	. . . . 5 ⊢ (( ≤ ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
30	28, 29	mp1i 13	. . . 4 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
31	6, 7, 10, 11	ordtprsuni 33865	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → 𝐵 = ∪ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))
32	4, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = ∪ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))
33	32, 17	eqeltrrd 2845	. . . . . . 7 ⊢ (𝜑 → ∪ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ∈ V)
34		uniexb 7799	. . . . . . 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)
35	33, 34	sylibr 234	. . . . . 6 ⊢ (𝜑 → ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ∈ V)
36		restval 17486	. . . . . 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)))
37	35, 18, 36	syl2anc 583	. . . . 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)))
38		sseqin2 4244	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
39	5, 38	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴)
40		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ( ≤ ∩ (𝐴 × 𝐴))
41	40	ordttopon 23222	. . . . . . . . . . . . . 14 ⊢ (( ≤ ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))))
42	28, 41	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))))
43	6, 7	prsssdm 33863	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
44	4, 5, 43	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
45	44	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ (𝜑 → (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
46	42, 45	eleqtrd 2846	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
47		toponmax 22953	. . . . . . . . . . . 12 ⊢ ((ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
48	46, 47	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
49	39, 48	eqeltrd 2844	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
50		elsni 4665	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ {𝐵} → 𝑣 = 𝐵)
51	50	ineq1d 4240	. . . . . . . . . . 11 ⊢ (𝑣 ∈ {𝐵} → (𝑣 ∩ 𝐴) = (𝐵 ∩ 𝐴))
52	51	eleq1d 2829	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝐵} → ((𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ (𝐵 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
53	49, 52	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ {𝐵} → (𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
54	53	ralrimiv 3151	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ {𝐵} (𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
55		ordtrest2NEW.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴)
56	6, 7, 1, 5, 55	ordtrest2NEWlem 33868	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
57		eqid 2740	. . . . . . . . . . . 12 ⊢ (ODual‘𝐾) = (ODual‘𝐾)
58	57, 6	odubas 18361	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘(ODual‘𝐾))
59	7	cnveqi 5899	. . . . . . . . . . . 12 ⊢ ◡ ≤ = ◡((le‘𝐾) ∩ (𝐵 × 𝐵))
60		cnvin 6176	. . . . . . . . . . . . 13 ⊢ ◡((le‘𝐾) ∩ (𝐵 × 𝐵)) = (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵))
61		cnvxp 6188	. . . . . . . . . . . . . 14 ⊢ ◡(𝐵 × 𝐵) = (𝐵 × 𝐵)
62	61	ineq2i 4238	. . . . . . . . . . . . 13 ⊢ (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵)) = (◡(le‘𝐾) ∩ (𝐵 × 𝐵))
63		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (le‘𝐾) = (le‘𝐾)
64	57, 63	oduleval 18359	. . . . . . . . . . . . . 14 ⊢ ◡(le‘𝐾) = (le‘(ODual‘𝐾))
65	64	ineq1i 4237	. . . . . . . . . . . . 13 ⊢ (◡(le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
66	60, 62, 65	3eqtri 2772	. . . . . . . . . . . 12 ⊢ ◡((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
67	59, 66	eqtri 2768	. . . . . . . . . . 11 ⊢ ◡ ≤ = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
68	57	odutos 32941	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Toset → (ODual‘𝐾) ∈ Toset)
69	1, 68	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ODual‘𝐾) ∈ Toset)
70		vex 3492	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
71		vex 3492	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
72	70, 71	brcnv 5907	. . . . . . . . . . . . . . 15 ⊢ (𝑦◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦)
73		vex 3492	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
74	71, 73	brcnv 5907	. . . . . . . . . . . . . . 15 ⊢ (𝑧◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧)
75	72, 74	anbi12ci 628	. . . . . . . . . . . . . 14 ⊢ ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))
76	75	rabbii 3449	. . . . . . . . . . . . 13 ⊢ {𝑧 ∈ 𝐵 ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} = {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}
77	76, 55	eqsstrid 4057	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴)
78	77	ancom2s 649	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴)
79	58, 67, 69, 5, 78	ordtrest2NEWlem 33868	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))))
80		vex 3492	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑤 ∈ V
81	80, 71	brcnv 5907	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑤)
82	81	bicomi 224	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ≤ 𝑤 ↔ 𝑤◡ ≤ 𝑧)
83	82	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ 𝑤◡ ≤ 𝑧))
84	83	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ 𝑧 ≤ 𝑤 ↔ ¬ 𝑤◡ ≤ 𝑧))
85	84	rabbidv 3451	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤} = {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧})
86	85	mpteq2dv 5268	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}) = (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧}))
87	86	rneqd 5963	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}) = ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧}))
88	6	ressprs 32936	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Proset )
89	4, 5, 88	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾 ↾s 𝐴) ∈ Proset )
90		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴))
91		eqid 2740	. . . . . . . . . . . . . . . 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))))
92	90, 91	ordtcnvNEW 33866	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ↾s 𝐴) ∈ Proset → (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))))))
93	89, 92	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))))))
94	6, 7	prsss 33862	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
95	4, 5, 94	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
96		eqid 2740	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴)
97	96, 63	ressle 17439	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
98	18, 97	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
99	96, 6	ressbas2 17296	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
100	5, 99	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
101	100	sqxpeqd 5732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 × 𝐴) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
102	98, 101	ineq12d 4242	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
103	95, 102	eqtrd 2780	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
104	103	cnveqd 5900	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ◡( ≤ ∩ (𝐴 × 𝐴)) = ◡((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
105	104	fveq2d 6924	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ordTop‘◡( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘◡((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))))
106	103	fveq2d 6924	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))))
107	93, 105, 106	3eqtr4d 2790	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘◡( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
108		cnvin 6176	. . . . . . . . . . . . . . 15 ⊢ ◡( ≤ ∩ (𝐴 × 𝐴)) = (◡ ≤ ∩ ◡(𝐴 × 𝐴))
109		cnvxp 6188	. . . . . . . . . . . . . . . 16 ⊢ ◡(𝐴 × 𝐴) = (𝐴 × 𝐴)
110	109	ineq2i 4238	. . . . . . . . . . . . . . 15 ⊢ (◡ ≤ ∩ ◡(𝐴 × 𝐴)) = (◡ ≤ ∩ (𝐴 × 𝐴))
111	108, 110	eqtri 2768	. . . . . . . . . . . . . 14 ⊢ ◡( ≤ ∩ (𝐴 × 𝐴)) = (◡ ≤ ∩ (𝐴 × 𝐴))
112	111	fveq2i 6923	. . . . . . . . . . . . 13 ⊢ (ordTop‘◡( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴)))
113	107, 112	eqtr3di 2795	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))))
114	113	eleq2d 2830	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∩ 𝐴) ∈ (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴)))))
115	87, 114	raleqbidv 3354	. . . . . . . . . 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)))))
116	79, 115	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
117		ralunb 4220	. . . . . . . . 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)))))
118	56, 116, 117	sylanbrc 582	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
119		ralunb 4220	. . . . . . . 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)))))
120	54, 118, 119	sylanbrc 582	. . . . . . 7 ⊢ (𝜑 → ∀𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
121		eqid 2740	. . . . . . . 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))
122	121	fmpt 7144	. . . . . . 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))))
123	120, 122	sylib 218	. . . . . 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))))
124	123	frnd 6755	. . . . 5 ⊢ (𝜑 → ran (𝑣 ∈ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↦ (𝑣 ∩ 𝐴)) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
125	37, 124	eqsstrd 4047	. . . 4 ⊢ (𝜑 → (({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
126		tgfiss 23019	. . . 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))))
127	30, 125, 126	syl2anc 583	. . 3 ⊢ (𝜑 → (topGen‘(fi‘(({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
128	24, 127	eqsstrd 4047	. 2 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾t 𝐴) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
129	9, 128	eqssd 4026	1 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾t 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  {csn 4648  ∪ cuni 4931   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698  ^◡ccnv 5699  dom cdm 5700  ran crn 5701  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ficfi 9479  Basecbs 17258   ↾_s cress 17287  lecple 17318   ↾_t crest 17480  topGenctg 17497  ordTopcordt 17559  ODualcodu 18356   Proset cproset 18363  Posetcpo 18377  Tosetctos 18486  Topctop 22920  TopOnctopon 22937  TopBasesctb 22973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-dec 12759  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-ple 17331  df-rest 17482  df-topgen 17503  df-ordt 17561  df-odu 18357  df-proset 18365  df-poset 18383  df-toset 18487  df-top 22921  df-topon 22938  df-bases 22974

This theorem is referenced by: (None)