lecldbas - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lecldbas		Structured version Visualization version GIF version

Theorem lecldbas 22723

Description: The set of closed intervals forms a closed subbasis for the topology on the extended reals. Since our definition of a basis is in terms of open sets, we express this by showing that the complements of closed intervals form an open subbasis for the topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Hypothesis**
Ref	Expression
lecldbas.1	⊢ 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ^* ∖ 𝑥))

**Assertion**
Ref	Expression
lecldbas	⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))

Proof of Theorem lecldbas Dummy variables 𝑎 𝑏 𝑐 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2733	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
2		eqid 2733	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
3	1, 2	leordtval2 22716	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)))))
4		fvex 6905	. . . 4 ⊢ (fi‘ran 𝐹) ∈ V
5		fvex 6905	. . . . . 6 ⊢ (ordTop‘ ≤ ) ∈ V
6		lecldbas.1	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ^* ∖ 𝑥))
7		iccf 13425	. . . . . . . . . . 11 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
8		ffn 6718	. . . . . . . . . . 11 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → [,] Fn (ℝ^* × ℝ^*))
9	7, 8	ax-mp 5	. . . . . . . . . 10 ⊢ [,] Fn (ℝ^* × ℝ^*)
10		ovelrn 7583	. . . . . . . . . 10 ⊢ ([,] Fn (ℝ^* × ℝ^) → (𝑥 ∈ ran [,] ↔ ∃𝑎 ∈ ℝ^ ∃𝑏 ∈ ℝ^* 𝑥 = (𝑎[,]𝑏)))
11	9, 10	ax-mp 5	. . . . . . . . 9 ⊢ (𝑥 ∈ ran [,] ↔ ∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* 𝑥 = (𝑎[,]𝑏))
12		difeq2 4117	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (𝑎[,]𝑏)))
13		iccordt 22718	. . . . . . . . . . . . 13 ⊢ (𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ ))
14		letopuni 22711	. . . . . . . . . . . . . 14 ⊢ ℝ^* = ∪ (ordTop‘ ≤ )
15	14	cldopn 22535	. . . . . . . . . . . . 13 ⊢ ((𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ )) → (ℝ^* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ ))
16	13, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℝ^* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ )
17	12, 16	eqeltrdi 2842	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
18	17	rexlimivw 3152	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℝ^* 𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
19	18	rexlimivw 3152	. . . . . . . . 9 ⊢ (∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* 𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
20	11, 19	sylbi 216	. . . . . . . 8 ⊢ (𝑥 ∈ ran [,] → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
21	6, 20	fmpti 7112	. . . . . . 7 ⊢ 𝐹:ran [,]⟶(ordTop‘ ≤ )
22		frn 6725	. . . . . . 7 ⊢ (𝐹:ran [,]⟶(ordTop‘ ≤ ) → ran 𝐹 ⊆ (ordTop‘ ≤ ))
23	21, 22	ax-mp 5	. . . . . 6 ⊢ ran 𝐹 ⊆ (ordTop‘ ≤ )
24	5, 23	ssexi 5323	. . . . 5 ⊢ ran 𝐹 ∈ V
25		eqid 2733	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
26		mnfxr 11271	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
27		fnovrn 7582	. . . . . . . . . . 11 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ 𝑦 ∈ ℝ^*) → (-∞[,]𝑦) ∈ ran [,])
28	9, 26, 27	mp3an12 1452	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (-∞[,]𝑦) ∈ ran [,])
29	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → -∞ ∈ ℝ^*)
30		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ∈ ℝ^*)
31		pnfxr 11268	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
32	31	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → +∞ ∈ ℝ^*)
33		mnfle 13114	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → -∞ ≤ 𝑦)
34		pnfge 13110	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
35		df-icc 13331	. . . . . . . . . . . . . . 15 ⊢ [,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 ≤ 𝑏)})
36		df-ioc 13329	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
37		xrltnle 11281	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (𝑦 < 𝑧 ↔ ¬ 𝑧 ≤ 𝑦))
38		xrletr 13137	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑧 ≤ 𝑦 ∧ 𝑦 ≤ +∞) → 𝑧 ≤ +∞))
39		xrlelttr 13135	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((-∞ ≤ 𝑦 ∧ 𝑦 < 𝑧) → -∞ < 𝑧))
40		xrltle 13128	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (-∞ < 𝑧 → -∞ ≤ 𝑧))
41	40	3adant2 1132	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (-∞ < 𝑧 → -∞ ≤ 𝑧))
42	39, 41	syld 47	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((-∞ ≤ 𝑦 ∧ 𝑦 < 𝑧) → -∞ ≤ 𝑧))
43	35, 36, 37, 35, 38, 42	ixxun 13340	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ 𝑦 ∧ 𝑦 ≤ +∞)) → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
44	29, 30, 32, 33, 34, 43	syl32anc 1379	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
45		iccmax 13400	. . . . . . . . . . . . 13 ⊢ (-∞[,]+∞) = ℝ^*
46	44, 45	eqtrdi 2789	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^*)
47		iccssxr 13407	. . . . . . . . . . . . 13 ⊢ (-∞[,]𝑦) ⊆ ℝ^*
48	35, 36, 37	ixxdisj 13339	. . . . . . . . . . . . . 14 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
49	26, 31, 48	mp3an13 1453	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
50		uneqdifeq 4493	. . . . . . . . . . . . 13 ⊢ (((-∞[,]𝑦) ⊆ ℝ^* ∧ ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅) → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^* ↔ (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
51	47, 49, 50	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^* ↔ (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
52	46, 51	mpbid 231	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞))
53	52	eqcomd 2739	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (𝑦(,]+∞) = (ℝ^* ∖ (-∞[,]𝑦)))
54		difeq2 4117	. . . . . . . . . . 11 ⊢ (𝑥 = (-∞[,]𝑦) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (-∞[,]𝑦)))
55	54	rspceeqv 3634	. . . . . . . . . 10 ⊢ (((-∞[,]𝑦) ∈ ran [,] ∧ (𝑦(,]+∞) = (ℝ^* ∖ (-∞[,]𝑦))) → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
56	28, 53, 55	syl2anc 585	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
57		xrex 12971	. . . . . . . . . . 11 ⊢ ℝ^* ∈ V
58	57	difexi 5329	. . . . . . . . . 10 ⊢ (ℝ^* ∖ 𝑥) ∈ V
59	6, 58	elrnmpti 5960	. . . . . . . . 9 ⊢ ((𝑦(,]+∞) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
60	56, 59	sylibr 233	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → (𝑦(,]+∞) ∈ ran 𝐹)
61	25, 60	fmpti 7112	. . . . . . 7 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)):ℝ^*⟶ran 𝐹
62		frn 6725	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)):ℝ^⟶ran 𝐹 → ran (𝑦 ∈ ℝ^ ↦ (𝑦(,]+∞)) ⊆ ran 𝐹)
63	61, 62	ax-mp 5	. . . . . 6 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ⊆ ran 𝐹
64		eqid 2733	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
65		fnovrn 7582	. . . . . . . . . . 11 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (𝑦[,]+∞) ∈ ran [,])
66	9, 31, 65	mp3an13 1453	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (𝑦[,]+∞) ∈ ran [,])
67		df-ico 13330	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 < 𝑏)})
68		xrlenlt 11279	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (𝑦 ≤ 𝑧 ↔ ¬ 𝑧 < 𝑦))
69		xrltletr 13136	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑧 < 𝑦 ∧ 𝑦 ≤ +∞) → 𝑧 < +∞))
70		xrltle 13128	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
71	70	3adant2 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
72	69, 71	syld 47	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑧 < 𝑦 ∧ 𝑦 ≤ +∞) → 𝑧 ≤ +∞))
73		xrletr 13137	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((-∞ ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → -∞ ≤ 𝑧))
74	67, 35, 68, 35, 72, 73	ixxun 13340	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ 𝑦 ∧ 𝑦 ≤ +∞)) → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
75	29, 30, 32, 33, 34, 74	syl32anc 1379	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
76		uncom 4154	. . . . . . . . . . . . 13 ⊢ ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = ((𝑦[,]+∞) ∪ (-∞[,)𝑦))
77	75, 76, 45	3eqtr3g 2796	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → ((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^*)
78		iccssxr 13407	. . . . . . . . . . . . 13 ⊢ (𝑦[,]+∞) ⊆ ℝ^*
79		incom 4202	. . . . . . . . . . . . . 14 ⊢ ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ((-∞[,)𝑦) ∩ (𝑦[,]+∞))
80	67, 35, 68	ixxdisj 13339	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
81	26, 31, 80	mp3an13 1453	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
82	79, 81	eqtrid 2785	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅)
83		uneqdifeq 4493	. . . . . . . . . . . . 13 ⊢ (((𝑦[,]+∞) ⊆ ℝ^* ∧ ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅) → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^* ↔ (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
84	78, 82, 83	sylancr 588	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^* ↔ (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
85	77, 84	mpbid 231	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦))
86	85	eqcomd 2739	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (-∞[,)𝑦) = (ℝ^* ∖ (𝑦[,]+∞)))
87		difeq2 4117	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦[,]+∞) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (𝑦[,]+∞)))
88	87	rspceeqv 3634	. . . . . . . . . 10 ⊢ (((𝑦[,]+∞) ∈ ran [,] ∧ (-∞[,)𝑦) = (ℝ^* ∖ (𝑦[,]+∞))) → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
89	66, 86, 88	syl2anc 585	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
90	6, 58	elrnmpti 5960	. . . . . . . . 9 ⊢ ((-∞[,)𝑦) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
91	89, 90	sylibr 233	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → (-∞[,)𝑦) ∈ ran 𝐹)
92	64, 91	fmpti 7112	. . . . . . 7 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)):ℝ^*⟶ran 𝐹
93		frn 6725	. . . . . . 7 ⊢ ((𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)):ℝ^⟶ran 𝐹 → ran (𝑦 ∈ ℝ^ ↦ (-∞[,)𝑦)) ⊆ ran 𝐹)
94	92, 93	ax-mp 5	. . . . . 6 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) ⊆ ran 𝐹
95	63, 94	unssi 4186	. . . . 5 ⊢ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ⊆ ran 𝐹
96		fiss 9419	. . . . 5 ⊢ ((ran 𝐹 ∈ V ∧ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ⊆ ran 𝐹) → (fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹))
97	24, 95, 96	mp2an 691	. . . 4 ⊢ (fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹)
98		tgss 22471	. . . 4 ⊢ (((fi‘ran 𝐹) ∈ V ∧ (fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹)) → (topGen‘(fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))))) ⊆ (topGen‘(fi‘ran 𝐹)))
99	4, 97, 98	mp2an 691	. . 3 ⊢ (topGen‘(fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))))) ⊆ (topGen‘(fi‘ran 𝐹))
100	3, 99	eqsstri 4017	. 2 ⊢ (ordTop‘ ≤ ) ⊆ (topGen‘(fi‘ran 𝐹))
101		letop 22710	. . 3 ⊢ (ordTop‘ ≤ ) ∈ Top
102		tgfiss 22494	. . 3 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ran 𝐹 ⊆ (ordTop‘ ≤ )) → (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ ))
103	101, 23, 102	mp2an 691	. 2 ⊢ (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ )
104	100, 103	eqssi 3999	1 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 Vcvv 3475 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 ran crn 5678 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ficfi 9405 +∞cpnf 11245 -∞cmnf 11246 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 (,]cioc 13325 [,)cico 13326 [,]cicc 13327 topGenctg 17383 ordTopcordt 17445 Topctop 22395 Clsdccld 22520

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-ioc 13329 df-ico 13330 df-icc 13331 df-topgen 17389 df-ordt 17447 df-ps 18519 df-tsr 18520 df-top 22396 df-topon 22413 df-bases 22449 df-cld 22523

This theorem is referenced by: (None)

W3C validator