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Theorem lecldbas 22476

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 2736	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
2		eqid 2736	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
3	1, 2	leordtval2 22469	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)))))
4		fvex 6838	. . . 4 ⊢ (fi‘ran 𝐹) ∈ V
5		fvex 6838	. . . . . 6 ⊢ (ordTop‘ ≤ ) ∈ V
6		lecldbas.1	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ^* ∖ 𝑥))
7		iccf 13281	. . . . . . . . . . 11 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
8		ffn 6651	. . . . . . . . . . 11 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → [,] Fn (ℝ^* × ℝ^*))
9	7, 8	ax-mp 5	. . . . . . . . . 10 ⊢ [,] Fn (ℝ^* × ℝ^*)
10		ovelrn 7510	. . . . . . . . . 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 4063	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (𝑎[,]𝑏)))
13		iccordt 22471	. . . . . . . . . . . . 13 ⊢ (𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ ))
14		letopuni 22464	. . . . . . . . . . . . . 14 ⊢ ℝ^* = ∪ (ordTop‘ ≤ )
15	14	cldopn 22288	. . . . . . . . . . . . 13 ⊢ ((𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ )) → (ℝ^* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ ))
16	13, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℝ^* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ )
17	12, 16	eqeltrdi 2845	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
18	17	rexlimivw 3144	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℝ^* 𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
19	18	rexlimivw 3144	. . . . . . . . 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 7042	. . . . . . 7 ⊢ 𝐹:ran [,]⟶(ordTop‘ ≤ )
22		frn 6658	. . . . . . 7 ⊢ (𝐹:ran [,]⟶(ordTop‘ ≤ ) → ran 𝐹 ⊆ (ordTop‘ ≤ ))
23	21, 22	ax-mp 5	. . . . . 6 ⊢ ran 𝐹 ⊆ (ordTop‘ ≤ )
24	5, 23	ssexi 5266	. . . . 5 ⊢ ran 𝐹 ∈ V
25		eqid 2736	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
26		mnfxr 11133	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
27		fnovrn 7509	. . . . . . . . . . 11 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ 𝑦 ∈ ℝ^*) → (-∞[,]𝑦) ∈ ran [,])
28	9, 26, 27	mp3an12 1450	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (-∞[,]𝑦) ∈ ran [,])
29	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → -∞ ∈ ℝ^*)
30		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ∈ ℝ^*)
31		pnfxr 11130	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
32	31	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → +∞ ∈ ℝ^*)
33		mnfle 12971	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → -∞ ≤ 𝑦)
34		pnfge 12967	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
35		df-icc 13187	. . . . . . . . . . . . . . 15 ⊢ [,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 ≤ 𝑏)})
36		df-ioc 13185	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
37		xrltnle 11143	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (𝑦 < 𝑧 ↔ ¬ 𝑧 ≤ 𝑦))
38		xrletr 12993	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑧 ≤ 𝑦 ∧ 𝑦 ≤ +∞) → 𝑧 ≤ +∞))
39		xrlelttr 12991	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((-∞ ≤ 𝑦 ∧ 𝑦 < 𝑧) → -∞ < 𝑧))
40		xrltle 12984	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (-∞ < 𝑧 → -∞ ≤ 𝑧))
41	40	3adant2 1130	. . . . . . . . . . . . . . . 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 13196	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ 𝑦 ∧ 𝑦 ≤ +∞)) → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
44	29, 30, 32, 33, 34, 43	syl32anc 1377	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
45		iccmax 13256	. . . . . . . . . . . . 13 ⊢ (-∞[,]+∞) = ℝ^*
46	44, 45	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^*)
47		iccssxr 13263	. . . . . . . . . . . . 13 ⊢ (-∞[,]𝑦) ⊆ ℝ^*
48	35, 36, 37	ixxdisj 13195	. . . . . . . . . . . . . 14 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
49	26, 31, 48	mp3an13 1451	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
50		uneqdifeq 4437	. . . . . . . . . . . . 13 ⊢ (((-∞[,]𝑦) ⊆ ℝ^* ∧ ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅) → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^* ↔ (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
51	47, 49, 50	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^* ↔ (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
52	46, 51	mpbid 231	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞))
53	52	eqcomd 2742	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (𝑦(,]+∞) = (ℝ^* ∖ (-∞[,]𝑦)))
54		difeq2 4063	. . . . . . . . . . 11 ⊢ (𝑥 = (-∞[,]𝑦) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (-∞[,]𝑦)))
55	54	rspceeqv 3584	. . . . . . . . . 10 ⊢ (((-∞[,]𝑦) ∈ ran [,] ∧ (𝑦(,]+∞) = (ℝ^* ∖ (-∞[,]𝑦))) → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
56	28, 53, 55	syl2anc 584	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
57		xrex 12828	. . . . . . . . . . 11 ⊢ ℝ^* ∈ V
58	57	difexi 5272	. . . . . . . . . 10 ⊢ (ℝ^* ∖ 𝑥) ∈ V
59	6, 58	elrnmpti 5901	. . . . . . . . 9 ⊢ ((𝑦(,]+∞) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
60	56, 59	sylibr 233	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → (𝑦(,]+∞) ∈ ran 𝐹)
61	25, 60	fmpti 7042	. . . . . . 7 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)):ℝ^*⟶ran 𝐹
62		frn 6658	. . . . . . 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 2736	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
65		fnovrn 7509	. . . . . . . . . . 11 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (𝑦[,]+∞) ∈ ran [,])
66	9, 31, 65	mp3an13 1451	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (𝑦[,]+∞) ∈ ran [,])
67		df-ico 13186	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 < 𝑏)})
68		xrlenlt 11141	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (𝑦 ≤ 𝑧 ↔ ¬ 𝑧 < 𝑦))
69		xrltletr 12992	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑧 < 𝑦 ∧ 𝑦 ≤ +∞) → 𝑧 < +∞))
70		xrltle 12984	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
71	70	3adant2 1130	. . . . . . . . . . . . . . . 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 12993	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((-∞ ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → -∞ ≤ 𝑧))
74	67, 35, 68, 35, 72, 73	ixxun 13196	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ 𝑦 ∧ 𝑦 ≤ +∞)) → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
75	29, 30, 32, 33, 34, 74	syl32anc 1377	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
76		uncom 4100	. . . . . . . . . . . . 13 ⊢ ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = ((𝑦[,]+∞) ∪ (-∞[,)𝑦))
77	75, 76, 45	3eqtr3g 2799	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → ((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^*)
78		iccssxr 13263	. . . . . . . . . . . . 13 ⊢ (𝑦[,]+∞) ⊆ ℝ^*
79		incom 4148	. . . . . . . . . . . . . 14 ⊢ ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ((-∞[,)𝑦) ∩ (𝑦[,]+∞))
80	67, 35, 68	ixxdisj 13195	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
81	26, 31, 80	mp3an13 1451	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
82	79, 81	eqtrid 2788	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅)
83		uneqdifeq 4437	. . . . . . . . . . . . 13 ⊢ (((𝑦[,]+∞) ⊆ ℝ^* ∧ ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅) → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^* ↔ (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
84	78, 82, 83	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^* ↔ (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
85	77, 84	mpbid 231	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦))
86	85	eqcomd 2742	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (-∞[,)𝑦) = (ℝ^* ∖ (𝑦[,]+∞)))
87		difeq2 4063	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦[,]+∞) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (𝑦[,]+∞)))
88	87	rspceeqv 3584	. . . . . . . . . 10 ⊢ (((𝑦[,]+∞) ∈ ran [,] ∧ (-∞[,)𝑦) = (ℝ^* ∖ (𝑦[,]+∞))) → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
89	66, 86, 88	syl2anc 584	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
90	6, 58	elrnmpti 5901	. . . . . . . . 9 ⊢ ((-∞[,)𝑦) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
91	89, 90	sylibr 233	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → (-∞[,)𝑦) ∈ ran 𝐹)
92	64, 91	fmpti 7042	. . . . . . 7 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)):ℝ^*⟶ran 𝐹
93		frn 6658	. . . . . . 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 4132	. . . . 5 ⊢ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ⊆ ran 𝐹
96		fiss 9281	. . . . 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 689	. . . 4 ⊢ (fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)))) ⊆ (fi‘ran 𝐹)
98		tgss 22224	. . . 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 689	. . 3 ⊢ (topGen‘(fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))))) ⊆ (topGen‘(fi‘ran 𝐹))
100	3, 99	eqsstri 3966	. 2 ⊢ (ordTop‘ ≤ ) ⊆ (topGen‘(fi‘ran 𝐹))
101		letop 22463	. . 3 ⊢ (ordTop‘ ≤ ) ∈ Top
102		tgfiss 22247	. . 3 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ran 𝐹 ⊆ (ordTop‘ ≤ )) → (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ ))
103	101, 23, 102	mp2an 689	. 2 ⊢ (topGen‘(fi‘ran 𝐹)) ⊆ (ordTop‘ ≤ )
104	100, 103	eqssi 3948	1 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∃wrex 3070 Vcvv 3441 ∖ cdif 3895 ∪ cun 3896 ∩ cin 3897 ⊆ wss 3898 ∅c0 4269 𝒫 cpw 4547 class class class wbr 5092 ↦ cmpt 5175 × cxp 5618 ran crn 5621 Fn wfn 6474 ⟶wf 6475 ‘cfv 6479 (class class class)co 7337 ficfi 9267 +∞cpnf 11107 -∞cmnf 11108 ℝ^*cxr 11109 < clt 11110 ≤ cle 11111 (,]cioc 13181 [,)cico 13182 [,]cicc 13183 topGenctg 17245 ordTopcordt 17307 Topctop 22148 Clsdccld 22273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-1o 8367 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fi 9268 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-ioc 13185 df-ico 13186 df-icc 13187 df-topgen 17251 df-ordt 17309 df-ps 18381 df-tsr 18382 df-top 22149 df-topon 22166 df-bases 22202 df-cld 22276

This theorem is referenced by: (None)

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