Theorem ordtrest2NEW 34336

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 18491	. . . 4 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset)
3		posprs 18389	. . . 4 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset )
4	1, 2, 3	3syl 19	. . 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 34334	. . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
9	4, 5, 8	syl2anc 596	. 2 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘ ≤ ) ↾t 𝐴))
10		eqid 2765	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) = ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})
11		eqid 2765	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}) = ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})
12	6, 7, 10, 11	ordtprsval 34331	. . . . . . 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 18	. . . . . 6 ⊢ (𝜑 → (ordTop‘ ≤ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))))
14	13	oveq1d 7431	. . . . 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 23163	. . . . . 6 ⊢ (fi‘({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})))) ∈ TopBases
16	6	fvexi 6899	. . . . . . . 8 ⊢ 𝐵 ∈ V
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
18	17, 5	ssexd 5297	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
19		tgrest 23345	. . . . . 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 599	. . . . 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 2803	. . . 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 17502	. . . . 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 6888	. . . 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 2818	. . 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 6898	. . . . . . . 8 ⊢ (le‘𝐾) ∈ V
26	25	inex1 5288	. . . . . . 7 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
27	7, 26	eqeltri 2861	. . . . . 6 ⊢ ≤ ∈ V
28	27	inex1 5288	. . . . 5 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) ∈ V
29		ordttop 23386	. . . . 5 ⊢ (( ≤ ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
30	28, 29	mp1i 14	. . . 4 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
31	6, 7, 10, 11	ordtprsuni 34332	. . . . . . . . 9 ⊢ (𝐾 ∈ Proset → 𝐵 = ∪ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))
32	4, 31	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = ∪ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))))
33	32, 17	eqeltrrd 2866	. . . . . . 7 ⊢ (𝜑 → ∪ ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ∈ V)
34		uniexb 7765	. . . . . . 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 237	. . . . . 6 ⊢ (𝜑 → ({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ∈ V)
36		restval 17496	. . . . . 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 596	. . . . 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 4176	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
39	5, 38	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴)
40		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ( ≤ ∩ (𝐴 × 𝐴))
41	40	ordttopon 23379	. . . . . . . . . . . . . 14 ⊢ (( ≤ ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))))
42	28, 41	mp1i 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))))
43	6, 7	prsssdm 34330	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
44	4, 5, 43	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
45	44	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (𝜑 → (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
46	42, 45	eleqtrd 2867	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
47		toponmax 23112	. . . . . . . . . . . 12 ⊢ ((ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
48	46, 47	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
49	39, 48	eqeltrd 2865	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
50		elsni 4608	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ {𝐵} → 𝑣 = 𝐵)
51	50	ineq1d 4172	. . . . . . . . . . 11 ⊢ (𝑣 ∈ {𝐵} → (𝑣 ∩ 𝐴) = (𝐵 ∩ 𝐴))
52	51	eleq1d 2850	. . . . . . . . . 10 ⊢ (𝑣 ∈ {𝐵} → ((𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ (𝐵 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
53	49, 52	syl5ibrcom 250	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ {𝐵} → (𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
54	53	ralrimiv 3158	. . . . . . . 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 34335	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
57		eqid 2765	. . . . . . . . . . . 12 ⊢ (ODual‘𝐾) = (ODual‘𝐾)
58	57, 6	odubas 18364	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘(ODual‘𝐾))
59	7	cnveqi 5862	. . . . . . . . . . . 12 ⊢ ◡ ≤ = ◡((le‘𝐾) ∩ (𝐵 × 𝐵))
60		cnvin 6143	. . . . . . . . . . . . 13 ⊢ ◡((le‘𝐾) ∩ (𝐵 × 𝐵)) = (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵))
61		cnvxp 6156	. . . . . . . . . . . . . 14 ⊢ ◡(𝐵 × 𝐵) = (𝐵 × 𝐵)
62	61	ineq2i 4170	. . . . . . . . . . . . 13 ⊢ (◡(le‘𝐾) ∩ ◡(𝐵 × 𝐵)) = (◡(le‘𝐾) ∩ (𝐵 × 𝐵))
63		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (le‘𝐾) = (le‘𝐾)
64	57, 63	oduleval 18362	. . . . . . . . . . . . . 14 ⊢ ◡(le‘𝐾) = (le‘(ODual‘𝐾))
65	64	ineq1i 4169	. . . . . . . . . . . . 13 ⊢ (◡(le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
66	60, 62, 65	3eqtri 2792	. . . . . . . . . . . 12 ⊢ ◡((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
67	59, 66	eqtri 2788	. . . . . . . . . . 11 ⊢ ◡ ≤ = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
68	57	odutos 33311	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Toset → (ODual‘𝐾) ∈ Toset)
69	1, 68	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (ODual‘𝐾) ∈ Toset)
70		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
71		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
72	70, 71	brcnv 5870	. . . . . . . . . . . . . . 15 ⊢ (𝑦◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑦)
73		vex 3461	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
74	71, 73	brcnv 5870	. . . . . . . . . . . . . . 15 ⊢ (𝑧◡ ≤ 𝑥 ↔ 𝑥 ≤ 𝑧)
75	72, 74	anbi12ci 641	. . . . . . . . . . . . . 14 ⊢ ((𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥) ↔ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))
76	75	rabbii 3423	. . . . . . . . . . . . 13 ⊢ {𝑧 ∈ 𝐵 ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} = {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}
77	76, 55	eqsstrid 3976	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴)
78	77	ancom2s 663	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑦◡ ≤ 𝑧 ∧ 𝑧◡ ≤ 𝑥)} ⊆ 𝐴)
79	58, 67, 69, 5, 78	ordtrest2NEWlem 34335	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))))
80		vex 3461	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑤 ∈ V
81	80, 71	brcnv 5870	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤◡ ≤ 𝑧 ↔ 𝑧 ≤ 𝑤)
82	81	bicomi 227	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ≤ 𝑤 ↔ 𝑤◡ ≤ 𝑧)
83	82	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ≤ 𝑤 ↔ 𝑤◡ ≤ 𝑧))
84	83	notbid 321	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (¬ 𝑧 ≤ 𝑤 ↔ ¬ 𝑤◡ ≤ 𝑧))
85	84	rabbidv 3425	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤} = {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧})
86	85	mpteq2dv 5207	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}) = (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧}))
87	86	rneqd 5930	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}) = ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤◡ ≤ 𝑧}))
88	6	ressprs 33309	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Proset )
89	4, 5, 88	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐾 ↾s 𝐴) ∈ Proset )
90		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴))
91		eqid 2765	. . . . . . . . . . . . . . . 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 34333	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 34329	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
95	4, 5, 94	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴)))
96		eqid 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴)
97	96, 63	ressle 17450	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
98	18, 97	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴)))
99	96, 6	ressbas2 17315	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
100	5, 99	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 = (Base‘(𝐾 ↾s 𝐴)))
101	100	sqxpeqd 5695	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐴 × 𝐴) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
102	98, 101	ineq12d 4174	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
103	95, 102	eqtrd 2800	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
104	103	cnveqd 5863	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ◡( ≤ ∩ (𝐴 × 𝐴)) = ◡((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
105	104	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ordTop‘◡( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘◡((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))))
106	103	fveq2d 6889	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))))
107	93, 105, 106	3eqtr4d 2810	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ordTop‘◡( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
108		cnvin 6143	. . . . . . . . . . . . . . 15 ⊢ ◡( ≤ ∩ (𝐴 × 𝐴)) = (◡ ≤ ∩ ◡(𝐴 × 𝐴))
109		cnvxp 6156	. . . . . . . . . . . . . . . 16 ⊢ ◡(𝐴 × 𝐴) = (𝐴 × 𝐴)
110	109	ineq2i 4170	. . . . . . . . . . . . . . 15 ⊢ (◡ ≤ ∩ ◡(𝐴 × 𝐴)) = (◡ ≤ ∩ (𝐴 × 𝐴))
111	108, 110	eqtri 2788	. . . . . . . . . . . . . 14 ⊢ ◡( ≤ ∩ (𝐴 × 𝐴)) = (◡ ≤ ∩ (𝐴 × 𝐴))
112	111	fveq2i 6888	. . . . . . . . . . . . 13 ⊢ (ordTop‘◡( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴)))
113	107, 112	eqtr3di 2815	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴))))
114	113	eleq2d 2851	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∩ 𝐴) ∈ (ordTop‘(◡ ≤ ∩ (𝐴 × 𝐴)))))
115	87, 114	raleqbidv 3340	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
117		ralunb 4150	. . . . . . . . 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 595	. . . . . . . 8 ⊢ (𝜑 → ∀𝑣 ∈ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
119		ralunb 4150	. . . . . . . 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 595	. . . . . . 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 2765	. . . . . . . 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 7109	. . . . . . 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 221	. . . . . 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 6718	. . . . 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 3972	. . . 4 ⊢ (𝜑 → (({𝐵} ∪ (ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) ∪ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑧 ≤ 𝑤}))) ↾t 𝐴) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
126		tgfiss 23177	. . . 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 596	. . 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 3972	. 2 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾t 𝐴) ⊆ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
129	9, 128	eqssd 3955	1 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾t 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418  Vcvv 3457   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  {csn 4591  ∪ cuni 4874   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661  ^◡ccnv 5662  dom cdm 5663  ran crn 5664  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416  ficfi 9373  Basecbs 17286   ↾_s cress 17307  lecple 17334   ↾_t crest 17490  topGenctg 17507  ordTopcordt 17570  ODualcodu 18359   Proset cproset 18365  Posetcpo 18380  Tosetctos 18487  Topctop 23079  TopOnctopon 23096  TopBasesctb 23131

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fi 9374  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-4 12316  df-5 12317  df-6 12318  df-7 12319  df-8 12320  df-9 12321  df-n0 12516  df-z 12603  df-dec 12723  df-sets 17241  df-slot 17259  df-ndx 17271  df-base 17287  df-ress 17308  df-ple 17347  df-rest 17492  df-topgen 17513  df-ordt 17572  df-odu 18360  df-proset 18367  df-poset 18386  df-toset 18488  df-top 23080  df-topon 23097  df-bases 23132

This theorem is used by: (None)