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Theorem tgcl 23178

Description: Show that a basis generates a topology. Remark in [Munkres] p. 79. (Contributed by NM, 17-Jul-2006.)

**Assertion**
Ref	Expression
tgcl	⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)

Proof of Theorem tgcl Dummy variables 𝑥 𝑦 𝑧 𝑢 𝑡 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniss 4882	. . . . . . . 8 ⊢ (𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ⊆ ∪ (topGen‘𝐵))
2	1	adantl 487	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ 𝑢 ⊆ ∪ (topGen‘𝐵))
3		unitg 23176	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵)
4	3	adantr 486	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ (topGen‘𝐵) = ∪ 𝐵)
5	2, 4	sseqtrd 3974	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ 𝑢 ⊆ ∪ 𝐵)
6		eluni2 4878	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑢 ↔ ∃𝑡 ∈ 𝑢 𝑥 ∈ 𝑡)
7		ssel2 3933	. . . . . . . . . . . 12 ⊢ ((𝑢 ⊆ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ (topGen‘𝐵))
8		eltg2b 23168	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ TopBases → (𝑡 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡)))
9		rsp 3255	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡) → (𝑥 ∈ 𝑡 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡)))
10	8, 9	biimtrdi 256	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ TopBases → (𝑡 ∈ (topGen‘𝐵) → (𝑥 ∈ 𝑡 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))))
11	10	imp31 423	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ TopBases ∧ 𝑡 ∈ (topGen‘𝐵)) ∧ 𝑥 ∈ 𝑡) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
12	11	an32s 665	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ TopBases ∧ 𝑥 ∈ 𝑡) ∧ 𝑡 ∈ (topGen‘𝐵)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
13	7, 12	sylan2 605	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑥 ∈ 𝑡) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑢)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
14	13	an42s 674	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑥 ∈ 𝑡)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
15		elssuni 4906	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑢 → 𝑡 ⊆ ∪ 𝑢)
16		sstr2 3945	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝑡 → (𝑡 ⊆ ∪ 𝑢 → 𝑦 ⊆ ∪ 𝑢))
17	15, 16	syl5com 32	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑢 → (𝑦 ⊆ 𝑡 → 𝑦 ⊆ ∪ 𝑢))
18	17	anim2d 624	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝑢 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡) → (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
19	18	reximdv 3182	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝑢 → (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
20	19	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑥 ∈ 𝑡)) → (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
21	14, 20	mpd 16	. . . . . . . . 9 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑥 ∈ 𝑡)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢))
22	21	rexlimdvaa 3169	. . . . . . . 8 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → (∃𝑡 ∈ 𝑢 𝑥 ∈ 𝑡 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
23	6, 22	biimtrid 245	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → (𝑥 ∈ ∪ 𝑢 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
24	23	ralrimiv 3158	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∀𝑥 ∈ ∪ 𝑢∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢))
25	5, 24	jca 521	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → (∪ 𝑢 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ ∪ 𝑢∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
26	25	ex 418	. . . 4 ⊢ (𝐵 ∈ TopBases → (𝑢 ⊆ (topGen‘𝐵) → (∪ 𝑢 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ ∪ 𝑢∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢))))
27		eltg2 23167	. . . 4 ⊢ (𝐵 ∈ TopBases → (∪ 𝑢 ∈ (topGen‘𝐵) ↔ (∪ 𝑢 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ ∪ 𝑢∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢))))
28	26, 27	sylibrd 262	. . 3 ⊢ (𝐵 ∈ TopBases → (𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ∈ (topGen‘𝐵)))
29	28	alrimiv 1960	. 2 ⊢ (𝐵 ∈ TopBases → ∀𝑢(𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ∈ (topGen‘𝐵)))
30		inss1 4189	. . . . . . . 8 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑢
31		tg1 23173	. . . . . . . 8 ⊢ (𝑢 ∈ (topGen‘𝐵) → 𝑢 ⊆ ∪ 𝐵)
32	30, 31	sstrid 3949	. . . . . . 7 ⊢ (𝑢 ∈ (topGen‘𝐵) → (𝑢 ∩ 𝑣) ⊆ ∪ 𝐵)
33	32	ad2antrl 741	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → (𝑢 ∩ 𝑣) ⊆ ∪ 𝐵)
34		eltg2 23167	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ TopBases → (𝑢 ∈ (topGen‘𝐵) ↔ (𝑢 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢))))
35	34	simplbda 505	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ∈ (topGen‘𝐵)) → ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢))
36		rsp 3255	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) → (𝑥 ∈ 𝑢 → ∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢)))
37	35, 36	syl 18	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ∈ (topGen‘𝐵)) → (𝑥 ∈ 𝑢 → ∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢)))
38		eltg2 23167	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ TopBases → (𝑣 ∈ (topGen‘𝐵) ↔ (𝑣 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑣 ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣))))
39	38	simplbda 505	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ TopBases ∧ 𝑣 ∈ (topGen‘𝐵)) → ∀𝑥 ∈ 𝑣 ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣))
40		rsp 3255	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑣 ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣) → (𝑥 ∈ 𝑣 → ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)))
41	39, 40	syl 18	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝑣 ∈ (topGen‘𝐵)) → (𝑥 ∈ 𝑣 → ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)))
42	37, 41	im2anan9 632	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ∈ (topGen‘𝐵)) ∧ (𝐵 ∈ TopBases ∧ 𝑣 ∈ (topGen‘𝐵))) → ((𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣) → (∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣))))
43		elin 3922	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑢 ∩ 𝑣) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))
44		reeanv 3239	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 ((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)) ↔ (∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)))
45	42, 43, 44	3imtr4g 299	. . . . . . . . 9 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ∈ (topGen‘𝐵)) ∧ (𝐵 ∈ TopBases ∧ 𝑣 ∈ (topGen‘𝐵))) → (𝑥 ∈ (𝑢 ∩ 𝑣) → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 ((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣))))
46	45	anandis 691	. . . . . . . 8 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → (𝑥 ∈ (𝑢 ∩ 𝑣) → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 ((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣))))
47		elin 3922	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑧 ∩ 𝑤) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤))
48	47	biimpri 231	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (𝑧 ∩ 𝑤))
49		ss2in 4197	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣) → (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣))
50	48, 49	anim12i 625	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤) ∧ (𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣)) → (𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)))
51	50	an4s 673	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)))
52		basis2 23160	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ TopBases ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ (𝑧 ∩ 𝑤))) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)))
53	52	adantllr 732	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ (𝑧 ∩ 𝑤))) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)))
54	53	adantrrr 738	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ (𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)))) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)))
55		sstr2 3945	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ⊆ (𝑧 ∩ 𝑤) → ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → 𝑡 ⊆ (𝑢 ∩ 𝑣)))
56	55	com12 33	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → (𝑡 ⊆ (𝑧 ∩ 𝑤) → 𝑡 ⊆ (𝑢 ∩ 𝑣)))
57	56	anim2d 624	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → ((𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)) → (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
58	57	reximdv 3182	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → (∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
59	58	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)) → (∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
60	59	ad2antll 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ (𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)))) → (∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
61	54, 60	mpd 16	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ (𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)))) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))
62	51, 61	sylanr2 696	. . . . . . . . . . . . 13 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ ((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)))) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))
63	62	rexlimdvaa 3169	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ TopBases ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) ∧ 𝑧 ∈ 𝐵) → (∃𝑤 ∈ 𝐵 ((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
64	63	rexlimdva 3168	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 ((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
65	64	ex 418	. . . . . . . . . 10 ⊢ (𝐵 ∈ TopBases → (𝑥 ∈ (𝑢 ∩ 𝑣) → (∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 ((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))))
66	65	a2d 30	. . . . . . . . 9 ⊢ (𝐵 ∈ TopBases → ((𝑥 ∈ (𝑢 ∩ 𝑣) → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 ((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣))) → (𝑥 ∈ (𝑢 ∩ 𝑣) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))))
67	66	imp 412	. . . . . . . 8 ⊢ ((𝐵 ∈ TopBases ∧ (𝑥 ∈ (𝑢 ∩ 𝑣) → ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 ((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)))) → (𝑥 ∈ (𝑢 ∩ 𝑣) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
68	46, 67	syldan 603	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → (𝑥 ∈ (𝑢 ∩ 𝑣) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
69	68	ralrimiv 3158	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → ∀𝑥 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))
70	33, 69	jca 521	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → ((𝑢 ∩ 𝑣) ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
71	70	ex 418	. . . 4 ⊢ (𝐵 ∈ TopBases → ((𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵)) → ((𝑢 ∩ 𝑣) ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))))
72		eltg2 23167	. . . 4 ⊢ (𝐵 ∈ TopBases → ((𝑢 ∩ 𝑣) ∈ (topGen‘𝐵) ↔ ((𝑢 ∩ 𝑣) ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))))
73	71, 72	sylibrd 262	. . 3 ⊢ (𝐵 ∈ TopBases → ((𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵)) → (𝑢 ∩ 𝑣) ∈ (topGen‘𝐵)))
74	73	ralrimivv 3208	. 2 ⊢ (𝐵 ∈ TopBases → ∀𝑢 ∈ (topGen‘𝐵)∀𝑣 ∈ (topGen‘𝐵)(𝑢 ∩ 𝑣) ∈ (topGen‘𝐵))
75		fvex 6898	. . 3 ⊢ (topGen‘𝐵) ∈ V
76		istopg 23104	. . 3 ⊢ ((topGen‘𝐵) ∈ V → ((topGen‘𝐵) ∈ Top ↔ (∀𝑢(𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ∈ (topGen‘𝐵)) ∧ ∀𝑢 ∈ (topGen‘𝐵)∀𝑣 ∈ (topGen‘𝐵)(𝑢 ∩ 𝑣) ∈ (topGen‘𝐵))))
77	75, 76	ax-mp 5	. 2 ⊢ ((topGen‘𝐵) ∈ Top ↔ (∀𝑢(𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ∈ (topGen‘𝐵)) ∧ ∀𝑢 ∈ (topGen‘𝐵)∀𝑣 ∈ (topGen‘𝐵)(𝑢 ∩ 𝑣) ∈ (topGen‘𝐵)))
78	29, 74, 77	sylanbrc 595	1 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∀wal 1568 = wceq 1570 ∈ wcel 2146 ∀wral 3081 ∃wrex 3091 Vcvv 3457 ∩ cin 3905 ⊆ wss 3906 ∪ cuni 4874 ‘cfv 6540 topGenctg 17514 Topctop 23102 TopBasesctb 23154

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 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 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-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-iota 6496 df-fun 6542 df-fv 6548 df-topgen 17520 df-top 23103 df-bases 23155

This theorem is used by: tgclb 23179 tgtopon 23180 bastop 23190 elcls3 23292 resttop 23369 leordtval2 23421 tgcmp 23610 2ndctop 23656 2ndcsb 23658 2ndcsep 23669 txtop 23779 pttop 23792 xkotop 23798 alexsubALT 24261 retop 24971 onsuctop 37003 kelac2lem 43851

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