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 23406

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 2765	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
2		eqid 2765	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
3	1, 2	leordtval2 23399	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)))))
4		fvex 6898	. . . 4 ⊢ (fi‘ran 𝐹) ∈ V
5		fvex 6898	. . . . . 6 ⊢ (ordTop‘ ≤ ) ∈ V
6		lecldbas.1	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ^* ∖ 𝑥))
7		iccf 13487	. . . . . . . . . . 11 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
8		ffn 6709	. . . . . . . . . . 11 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → [,] Fn (ℝ^* × ℝ^*))
9	7, 8	ax-mp 5	. . . . . . . . . 10 ⊢ [,] Fn (ℝ^* × ℝ^*)
10		ovelrn 7592	. . . . . . . . . 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 4075	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (𝑎[,]𝑏)))
13		iccordt 23401	. . . . . . . . . . . . 13 ⊢ (𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ ))
14		letopuni 23394	. . . . . . . . . . . . . 14 ⊢ ℝ^* = ∪ (ordTop‘ ≤ )
15	14	cldopn 23218	. . . . . . . . . . . . 13 ⊢ ((𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ )) → (ℝ^* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ ))
16	13, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℝ^* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ )
17	12, 16	eqeltrdi 2873	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
18	17	rexlimivw 3164	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℝ^* 𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
19	18	rexlimivw 3164	. . . . . . . . 9 ⊢ (∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* 𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
20	11, 19	sylbi 220	. . . . . . . 8 ⊢ (𝑥 ∈ ran [,] → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
21	6, 20	fmpti 7111	. . . . . . 7 ⊢ 𝐹:ran [,]⟶(ordTop‘ ≤ )
22		frn 6717	. . . . . . 7 ⊢ (𝐹:ran [,]⟶(ordTop‘ ≤ ) → ran 𝐹 ⊆ (ordTop‘ ≤ ))
23	21, 22	ax-mp 5	. . . . . 6 ⊢ ran 𝐹 ⊆ (ordTop‘ ≤ )
24	5, 23	ssexi 5295	. . . . 5 ⊢ ran 𝐹 ∈ V
25		eqid 2765	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
26		mnfxr 11277	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
27		fnovrn 7591	. . . . . . . . . . 11 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ 𝑦 ∈ ℝ^*) → (-∞[,]𝑦) ∈ ran [,])
28	9, 26, 27	mp3an12 1480	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (-∞[,]𝑦) ∈ ran [,])
29	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → -∞ ∈ ℝ^*)
30		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ∈ ℝ^*)
31		pnfxr 11274	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
32	31	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → +∞ ∈ ℝ^*)
33		mnfle 13172	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → -∞ ≤ 𝑦)
34		pnfge 13167	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
35		df-icc 13391	. . . . . . . . . . . . . . 15 ⊢ [,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 ≤ 𝑏)})
36		df-ioc 13389	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
37		xrltnle 11287	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (𝑦 < 𝑧 ↔ ¬ 𝑧 ≤ 𝑦))
38		xrletr 13195	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑧 ≤ 𝑦 ∧ 𝑦 ≤ +∞) → 𝑧 ≤ +∞))
39		xrlelttr 13193	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((-∞ ≤ 𝑦 ∧ 𝑦 < 𝑧) → -∞ < 𝑧))
40		xrltle 13186	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (-∞ < 𝑧 → -∞ ≤ 𝑧))
41	40	3adant2 1149	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (-∞ < 𝑧 → -∞ ≤ 𝑧))
42	39, 41	syld 48	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((-∞ ≤ 𝑦 ∧ 𝑦 < 𝑧) → -∞ ≤ 𝑧))
43	35, 36, 37, 35, 38, 42	ixxun 13400	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ 𝑦 ∧ 𝑦 ≤ +∞)) → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
44	29, 30, 32, 33, 34, 43	syl32anc 1405	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
45		iccmax 13462	. . . . . . . . . . . . 13 ⊢ (-∞[,]+∞) = ℝ^*
46	44, 45	eqtrdi 2816	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^*)
47		iccssxr 13469	. . . . . . . . . . . . 13 ⊢ (-∞[,]𝑦) ⊆ ℝ^*
48	35, 36, 37	ixxdisj 13399	. . . . . . . . . . . . . 14 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
49	26, 31, 48	mp3an13 1481	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
50		uneqdifeq 4455	. . . . . . . . . . . . 13 ⊢ (((-∞[,]𝑦) ⊆ ℝ^* ∧ ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅) → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^* ↔ (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
51	47, 49, 50	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^* ↔ (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
52	46, 51	mpbid 235	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞))
53	52	eqcomd 2771	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (𝑦(,]+∞) = (ℝ^* ∖ (-∞[,]𝑦)))
54		difeq2 4075	. . . . . . . . . . 11 ⊢ (𝑥 = (-∞[,]𝑦) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (-∞[,]𝑦)))
55	54	rspceeqv 3606	. . . . . . . . . 10 ⊢ (((-∞[,]𝑦) ∈ ran [,] ∧ (𝑦(,]+∞) = (ℝ^* ∖ (-∞[,]𝑦))) → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
56	28, 53, 55	syl2anc 596	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
57		xrex 13023	. . . . . . . . . . 11 ⊢ ℝ^* ∈ V
58	57	difexi 5303	. . . . . . . . . 10 ⊢ (ℝ^* ∖ 𝑥) ∈ V
59	6, 58	elrnmpti 5954	. . . . . . . . 9 ⊢ ((𝑦(,]+∞) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
60	56, 59	sylibr 237	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → (𝑦(,]+∞) ∈ ran 𝐹)
61	25, 60	fmpti 7111	. . . . . . 7 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)):ℝ^*⟶ran 𝐹
62		frn 6717	. . . . . . 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 2765	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
65		fnovrn 7591	. . . . . . . . . . 11 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (𝑦[,]+∞) ∈ ran [,])
66	9, 31, 65	mp3an13 1481	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (𝑦[,]+∞) ∈ ran [,])
67		df-ico 13390	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 < 𝑏)})
68		xrlenlt 11285	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (𝑦 ≤ 𝑧 ↔ ¬ 𝑧 < 𝑦))
69		xrltletr 13194	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑧 < 𝑦 ∧ 𝑦 ≤ +∞) → 𝑧 < +∞))
70		xrltle 13186	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
71	70	3adant2 1149	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
72	69, 71	syld 48	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑧 < 𝑦 ∧ 𝑦 ≤ +∞) → 𝑧 ≤ +∞))
73		xrletr 13195	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((-∞ ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → -∞ ≤ 𝑧))
74	67, 35, 68, 35, 72, 73	ixxun 13400	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ 𝑦 ∧ 𝑦 ≤ +∞)) → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
75	29, 30, 32, 33, 34, 74	syl32anc 1405	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
76		uncom 4112	. . . . . . . . . . . . 13 ⊢ ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = ((𝑦[,]+∞) ∪ (-∞[,)𝑦))
77	75, 76, 45	3eqtr3g 2823	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → ((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^*)
78		iccssxr 13469	. . . . . . . . . . . . 13 ⊢ (𝑦[,]+∞) ⊆ ℝ^*
79		incom 4162	. . . . . . . . . . . . . 14 ⊢ ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ((-∞[,)𝑦) ∩ (𝑦[,]+∞))
80	67, 35, 68	ixxdisj 13399	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
81	26, 31, 80	mp3an13 1481	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
82	79, 81	eqtrid 2812	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅)
83		uneqdifeq 4455	. . . . . . . . . . . . 13 ⊢ (((𝑦[,]+∞) ⊆ ℝ^* ∧ ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅) → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^* ↔ (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
84	78, 82, 83	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^* ↔ (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
85	77, 84	mpbid 235	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦))
86	85	eqcomd 2771	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (-∞[,)𝑦) = (ℝ^* ∖ (𝑦[,]+∞)))
87		difeq2 4075	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦[,]+∞) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (𝑦[,]+∞)))
88	87	rspceeqv 3606	. . . . . . . . . 10 ⊢ (((𝑦[,]+∞) ∈ ran [,] ∧ (-∞[,)𝑦) = (ℝ^* ∖ (𝑦[,]+∞))) → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
89	66, 86, 88	syl2anc 596	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
90	6, 58	elrnmpti 5954	. . . . . . . . 9 ⊢ ((-∞[,)𝑦) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
91	89, 90	sylibr 237	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → (-∞[,)𝑦) ∈ ran 𝐹)
92	64, 91	fmpti 7111	. . . . . . 7 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)):ℝ^*⟶ran 𝐹
93		frn 6717	. . . . . . 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 4144	. . . . 5 ⊢ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ⊆ ran 𝐹
96		fiss 9387	. . . . 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 705	. . . 4 ⊢ (fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹)
98		tgss 23155	. . . 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 705	. . 3 ⊢ (topGen‘(fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))))) ⊆ (topGen‘(fi‘ran 𝐹))
100	3, 99	eqsstri 3984	. 2 ⊢ (ordTop‘ ≤ ) ⊆ (topGen‘(fi‘ran 𝐹))
101		letop 23393	. . 3 ⊢ (ordTop‘ ≤ ) ∈ Top
102		tgfiss 23178	. . 3 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ran 𝐹 ⊆ (ordTop‘ ≤ )) → (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ ))
103	101, 23, 102	mp2an 705	. 2 ⊢ (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ )
104	100, 103	eqssi 3954	1 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∃wrex 3091 Vcvv 3457 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 𝒫 cpw 4564 class class class wbr 5111 ↦ cmpt 5194 × cxp 5661 ran crn 5664 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7416 ficfi 9373 +∞cpnf 11251 -∞cmnf 11252 ℝ^*cxr 11253 < clt 11254 ≤ cle 11255 (,]cioc 13385 [,)cico 13386 [,]cicc 13387 topGenctg 17508 ordTopcordt 17571 Topctop 23080 Clsdccld 23203

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-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-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-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-ioc 13389 df-ico 13390 df-icc 13391 df-topgen 17514 df-ordt 17573 df-ps 18640 df-tsr 18641 df-top 23081 df-topon 23098 df-bases 23133 df-cld 23206

This theorem is used by: (None)

W3C validator