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Theorem lecldbas 23248

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 2740	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
2		eqid 2740	. . . 4 ⊢ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
3	1, 2	leordtval2 23241	. . 3 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘(ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)))))
4		fvex 6933	. . . 4 ⊢ (fi‘ran 𝐹) ∈ V
5		fvex 6933	. . . . . 6 ⊢ (ordTop‘ ≤ ) ∈ V
6		lecldbas.1	. . . . . . . 8 ⊢ 𝐹 = (𝑥 ∈ ran [,] ↦ (ℝ^* ∖ 𝑥))
7		iccf 13508	. . . . . . . . . . 11 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
8		ffn 6747	. . . . . . . . . . 11 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → [,] Fn (ℝ^* × ℝ^*))
9	7, 8	ax-mp 5	. . . . . . . . . 10 ⊢ [,] Fn (ℝ^* × ℝ^*)
10		ovelrn 7626	. . . . . . . . . 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 4143	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (𝑎[,]𝑏)))
13		iccordt 23243	. . . . . . . . . . . . 13 ⊢ (𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ ))
14		letopuni 23236	. . . . . . . . . . . . . 14 ⊢ ℝ^* = ∪ (ordTop‘ ≤ )
15	14	cldopn 23060	. . . . . . . . . . . . 13 ⊢ ((𝑎[,]𝑏) ∈ (Clsd‘(ordTop‘ ≤ )) → (ℝ^* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ ))
16	13, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ (ℝ^* ∖ (𝑎[,]𝑏)) ∈ (ordTop‘ ≤ )
17	12, 16	eqeltrdi 2852	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
18	17	rexlimivw 3157	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℝ^* 𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
19	18	rexlimivw 3157	. . . . . . . . 9 ⊢ (∃𝑎 ∈ ℝ^* ∃𝑏 ∈ ℝ^* 𝑥 = (𝑎[,]𝑏) → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
20	11, 19	sylbi 217	. . . . . . . 8 ⊢ (𝑥 ∈ ran [,] → (ℝ^* ∖ 𝑥) ∈ (ordTop‘ ≤ ))
21	6, 20	fmpti 7146	. . . . . . 7 ⊢ 𝐹:ran [,]⟶(ordTop‘ ≤ )
22		frn 6754	. . . . . . 7 ⊢ (𝐹:ran [,]⟶(ordTop‘ ≤ ) → ran 𝐹 ⊆ (ordTop‘ ≤ ))
23	21, 22	ax-mp 5	. . . . . 6 ⊢ ran 𝐹 ⊆ (ordTop‘ ≤ )
24	5, 23	ssexi 5340	. . . . 5 ⊢ ran 𝐹 ∈ V
25		eqid 2740	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) = (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞))
26		mnfxr 11347	. . . . . . . . . . 11 ⊢ -∞ ∈ ℝ^*
27		fnovrn 7625	. . . . . . . . . . 11 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ -∞ ∈ ℝ^ ∧ 𝑦 ∈ ℝ^*) → (-∞[,]𝑦) ∈ ran [,])
28	9, 26, 27	mp3an12 1451	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (-∞[,]𝑦) ∈ ran [,])
29	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → -∞ ∈ ℝ^*)
30		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ∈ ℝ^*)
31		pnfxr 11344	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
32	31	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → +∞ ∈ ℝ^*)
33		mnfle 13197	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → -∞ ≤ 𝑦)
34		pnfge 13193	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
35		df-icc 13414	. . . . . . . . . . . . . . 15 ⊢ [,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 ≤ 𝑏)})
36		df-ioc 13412	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
37		xrltnle 11357	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (𝑦 < 𝑧 ↔ ¬ 𝑧 ≤ 𝑦))
38		xrletr 13220	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑧 ≤ 𝑦 ∧ 𝑦 ≤ +∞) → 𝑧 ≤ +∞))
39		xrlelttr 13218	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((-∞ ≤ 𝑦 ∧ 𝑦 < 𝑧) → -∞ < 𝑧))
40		xrltle 13211	. . . . . . . . . . . . . . . . 17 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (-∞ < 𝑧 → -∞ ≤ 𝑧))
41	40	3adant2 1131	. . . . . . . . . . . . . . . 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 13423	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ 𝑦 ∧ 𝑦 ≤ +∞)) → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
44	29, 30, 32, 33, 34, 43	syl32anc 1378	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = (-∞[,]+∞))
45		iccmax 13483	. . . . . . . . . . . . 13 ⊢ (-∞[,]+∞) = ℝ^*
46	44, 45	eqtrdi 2796	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^*)
47		iccssxr 13490	. . . . . . . . . . . . 13 ⊢ (-∞[,]𝑦) ⊆ ℝ^*
48	35, 36, 37	ixxdisj 13422	. . . . . . . . . . . . . 14 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
49	26, 31, 48	mp3an13 1452	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅)
50		uneqdifeq 4516	. . . . . . . . . . . . 13 ⊢ (((-∞[,]𝑦) ⊆ ℝ^* ∧ ((-∞[,]𝑦) ∩ (𝑦(,]+∞)) = ∅) → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^* ↔ (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
51	47, 49, 50	sylancr 586	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (((-∞[,]𝑦) ∪ (𝑦(,]+∞)) = ℝ^* ↔ (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞)))
52	46, 51	mpbid 232	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (ℝ^* ∖ (-∞[,]𝑦)) = (𝑦(,]+∞))
53	52	eqcomd 2746	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (𝑦(,]+∞) = (ℝ^* ∖ (-∞[,]𝑦)))
54		difeq2 4143	. . . . . . . . . . 11 ⊢ (𝑥 = (-∞[,]𝑦) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (-∞[,]𝑦)))
55	54	rspceeqv 3658	. . . . . . . . . 10 ⊢ (((-∞[,]𝑦) ∈ ran [,] ∧ (𝑦(,]+∞) = (ℝ^* ∖ (-∞[,]𝑦))) → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
56	28, 53, 55	syl2anc 583	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
57		xrex 13052	. . . . . . . . . . 11 ⊢ ℝ^* ∈ V
58	57	difexi 5348	. . . . . . . . . 10 ⊢ (ℝ^* ∖ 𝑥) ∈ V
59	6, 58	elrnmpti 5985	. . . . . . . . 9 ⊢ ((𝑦(,]+∞) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](𝑦(,]+∞) = (ℝ^* ∖ 𝑥))
60	56, 59	sylibr 234	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → (𝑦(,]+∞) ∈ ran 𝐹)
61	25, 60	fmpti 7146	. . . . . . 7 ⊢ (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)):ℝ^*⟶ran 𝐹
62		frn 6754	. . . . . . 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 2740	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)) = (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))
65		fnovrn 7625	. . . . . . . . . . 11 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ +∞ ∈ ℝ^*) → (𝑦[,]+∞) ∈ ran [,])
66	9, 31, 65	mp3an13 1452	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (𝑦[,]+∞) ∈ ran [,])
67		df-ico 13413	. . . . . . . . . . . . . . 15 ⊢ [,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 < 𝑏)})
68		xrlenlt 11355	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (𝑦 ≤ 𝑧 ↔ ¬ 𝑧 < 𝑦))
69		xrltletr 13219	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑧 < 𝑦 ∧ 𝑦 ≤ +∞) → 𝑧 < +∞))
70		xrltle 13211	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝑧 < +∞ → 𝑧 ≤ +∞))
71	70	3adant2 1131	. . . . . . . . . . . . . . . 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 13220	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((-∞ ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → -∞ ≤ 𝑧))
74	67, 35, 68, 35, 72, 73	ixxun 13423	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ ≤ 𝑦 ∧ 𝑦 ≤ +∞)) → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
75	29, 30, 32, 33, 34, 74	syl32anc 1378	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = (-∞[,]+∞))
76		uncom 4181	. . . . . . . . . . . . 13 ⊢ ((-∞[,)𝑦) ∪ (𝑦[,]+∞)) = ((𝑦[,]+∞) ∪ (-∞[,)𝑦))
77	75, 76, 45	3eqtr3g 2803	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → ((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^*)
78		iccssxr 13490	. . . . . . . . . . . . 13 ⊢ (𝑦[,]+∞) ⊆ ℝ^*
79		incom 4230	. . . . . . . . . . . . . 14 ⊢ ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ((-∞[,)𝑦) ∩ (𝑦[,]+∞))
80	67, 35, 68	ixxdisj 13422	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
81	26, 31, 80	mp3an13 1452	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ^* → ((-∞[,)𝑦) ∩ (𝑦[,]+∞)) = ∅)
82	79, 81	eqtrid 2792	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ^* → ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅)
83		uneqdifeq 4516	. . . . . . . . . . . . 13 ⊢ (((𝑦[,]+∞) ⊆ ℝ^* ∧ ((𝑦[,]+∞) ∩ (-∞[,)𝑦)) = ∅) → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^* ↔ (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
84	78, 82, 83	sylancr 586	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ^* → (((𝑦[,]+∞) ∪ (-∞[,)𝑦)) = ℝ^* ↔ (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦)))
85	77, 84	mpbid 232	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ^* → (ℝ^* ∖ (𝑦[,]+∞)) = (-∞[,)𝑦))
86	85	eqcomd 2746	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ^* → (-∞[,)𝑦) = (ℝ^* ∖ (𝑦[,]+∞)))
87		difeq2 4143	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦[,]+∞) → (ℝ^* ∖ 𝑥) = (ℝ^* ∖ (𝑦[,]+∞)))
88	87	rspceeqv 3658	. . . . . . . . . 10 ⊢ (((𝑦[,]+∞) ∈ ran [,] ∧ (-∞[,)𝑦) = (ℝ^* ∖ (𝑦[,]+∞))) → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
89	66, 86, 88	syl2anc 583	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ^* → ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
90	6, 58	elrnmpti 5985	. . . . . . . . 9 ⊢ ((-∞[,)𝑦) ∈ ran 𝐹 ↔ ∃𝑥 ∈ ran [,](-∞[,)𝑦) = (ℝ^* ∖ 𝑥))
91	89, 90	sylibr 234	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ^* → (-∞[,)𝑦) ∈ ran 𝐹)
92	64, 91	fmpti 7146	. . . . . . 7 ⊢ (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦)):ℝ^*⟶ran 𝐹
93		frn 6754	. . . . . . 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 4214	. . . . 5 ⊢ (ran (𝑦 ∈ ℝ^* ↦ (𝑦(,]+∞)) ∪ ran (𝑦 ∈ ℝ^* ↦ (-∞[,)𝑦))) ⊆ ran 𝐹
96		fiss 9493	. . . . 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 22996	. . . 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 4043	. 2 ⊢ (ordTop‘ ≤ ) ⊆ (topGen‘(fi‘ran 𝐹))
101		letop 23235	. . 3 ⊢ (ordTop‘ ≤ ) ∈ Top
102		tgfiss 23019	. . 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 4025	1 ⊢ (ordTop‘ ≤ ) = (topGen‘(fi‘ran 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∃wrex 3076 Vcvv 3488 ∖ cdif 3973 ∪ cun 3974 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 𝒫 cpw 4622 class class class wbr 5166 ↦ cmpt 5249 × cxp 5698 ran crn 5701 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ficfi 9479 +∞cpnf 11321 -∞cmnf 11322 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 (,]cioc 13408 [,)cico 13409 [,]cicc 13410 topGenctg 17497 ordTopcordt 17559 Topctop 22920 Clsdccld 23045

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-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-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-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-ioc 13412 df-ico 13413 df-icc 13414 df-topgen 17503 df-ordt 17561 df-ps 18636 df-tsr 18637 df-top 22921 df-topon 22938 df-bases 22974 df-cld 23048

This theorem is referenced by: (None)

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