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Theorem tgcl 22925

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 4873	. . . . . . . 8 ⊢ (𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ⊆ ∪ (topGen‘𝐵))
2	1	adantl 481	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ 𝑢 ⊆ ∪ (topGen‘𝐵))
3		unitg 22923	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵)
4	3	adantr 480	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ (topGen‘𝐵) = ∪ 𝐵)
5	2, 4	sseqtrd 3972	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ 𝑢 ⊆ ∪ 𝐵)
6		eluni2 4869	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑢 ↔ ∃𝑡 ∈ 𝑢 𝑥 ∈ 𝑡)
7		ssel2 3930	. . . . . . . . . . . 12 ⊢ ((𝑢 ⊆ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ (topGen‘𝐵))
8		eltg2b 22915	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ TopBases → (𝑡 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡)))
9		rsp 3226	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡) → (𝑥 ∈ 𝑡 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡)))
10	8, 9	biimtrdi 253	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ TopBases → (𝑡 ∈ (topGen‘𝐵) → (𝑥 ∈ 𝑡 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))))
11	10	imp31 417	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ TopBases ∧ 𝑡 ∈ (topGen‘𝐵)) ∧ 𝑥 ∈ 𝑡) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
12	11	an32s 653	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ TopBases ∧ 𝑥 ∈ 𝑡) ∧ 𝑡 ∈ (topGen‘𝐵)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
13	7, 12	sylan2 594	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑥 ∈ 𝑡) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑢)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
14	13	an42s 662	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑥 ∈ 𝑡)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
15		elssuni 4896	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑢 → 𝑡 ⊆ ∪ 𝑢)
16		sstr2 3942	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝑡 → (𝑡 ⊆ ∪ 𝑢 → 𝑦 ⊆ ∪ 𝑢))
17	15, 16	syl5com 31	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑢 → (𝑦 ⊆ 𝑡 → 𝑦 ⊆ ∪ 𝑢))
18	17	anim2d 613	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝑢 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡) → (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
19	18	reximdv 3153	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝑢 → (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
20	19	ad2antrl 729	. . . . . . . . . 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 15	. . . . . . . . 9 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑥 ∈ 𝑡)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢))
22	21	rexlimdvaa 3140	. . . . . . . 8 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → (∃𝑡 ∈ 𝑢 𝑥 ∈ 𝑡 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
23	6, 22	biimtrid 242	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → (𝑥 ∈ ∪ 𝑢 → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
24	23	ralrimiv 3129	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∀𝑥 ∈ ∪ 𝑢∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢))
25	5, 24	jca 511	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → (∪ 𝑢 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ ∪ 𝑢∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
26	25	ex 412	. . . 4 ⊢ (𝐵 ∈ TopBases → (𝑢 ⊆ (topGen‘𝐵) → (∪ 𝑢 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ ∪ 𝑢∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢))))
27		eltg2 22914	. . . 4 ⊢ (𝐵 ∈ TopBases → (∪ 𝑢 ∈ (topGen‘𝐵) ↔ (∪ 𝑢 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ ∪ 𝑢∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢))))
28	26, 27	sylibrd 259	. . 3 ⊢ (𝐵 ∈ TopBases → (𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ∈ (topGen‘𝐵)))
29	28	alrimiv 1929	. 2 ⊢ (𝐵 ∈ TopBases → ∀𝑢(𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ∈ (topGen‘𝐵)))
30		inss1 4191	. . . . . . . 8 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑢
31		tg1 22920	. . . . . . . 8 ⊢ (𝑢 ∈ (topGen‘𝐵) → 𝑢 ⊆ ∪ 𝐵)
32	30, 31	sstrid 3947	. . . . . . 7 ⊢ (𝑢 ∈ (topGen‘𝐵) → (𝑢 ∩ 𝑣) ⊆ ∪ 𝐵)
33	32	ad2antrl 729	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → (𝑢 ∩ 𝑣) ⊆ ∪ 𝐵)
34		eltg2 22914	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ TopBases → (𝑢 ∈ (topGen‘𝐵) ↔ (𝑢 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢))))
35	34	simplbda 499	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ∈ (topGen‘𝐵)) → ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢))
36		rsp 3226	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) → (𝑥 ∈ 𝑢 → ∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢)))
37	35, 36	syl 17	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ∈ (topGen‘𝐵)) → (𝑥 ∈ 𝑢 → ∃𝑧 ∈ 𝐵 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢)))
38		eltg2 22914	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ TopBases → (𝑣 ∈ (topGen‘𝐵) ↔ (𝑣 ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ 𝑣 ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣))))
39	38	simplbda 499	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ TopBases ∧ 𝑣 ∈ (topGen‘𝐵)) → ∀𝑥 ∈ 𝑣 ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣))
40		rsp 3226	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑣 ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣) → (𝑥 ∈ 𝑣 → ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)))
41	39, 40	syl 17	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ TopBases ∧ 𝑣 ∈ (topGen‘𝐵)) → (𝑥 ∈ 𝑣 → ∃𝑤 ∈ 𝐵 (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)))
42	37, 41	im2anan9 621	. . . . . . . . . 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 3919	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑢 ∩ 𝑣) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))
44		reeanv 3210	. . . . . . . . . 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 296	. . . . . . . . 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 679	. . . . . . . 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 3919	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑧 ∩ 𝑤) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤))
48	47	biimpri 228	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (𝑧 ∩ 𝑤))
49		ss2in 4199	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣) → (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣))
50	48, 49	anim12i 614	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤) ∧ (𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣)) → (𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)))
51	50	an4s 661	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)))
52		basis2 22907	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ TopBases ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ (𝑧 ∩ 𝑤))) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)))
53	52	adantllr 720	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ (𝑧 ∩ 𝑤))) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)))
54	53	adantrrr 726	. . . . . . . . . . . . . . 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 3942	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 ⊆ (𝑧 ∩ 𝑤) → ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → 𝑡 ⊆ (𝑢 ∩ 𝑣)))
56	55	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → (𝑡 ⊆ (𝑧 ∩ 𝑤) → 𝑡 ⊆ (𝑢 ∩ 𝑣)))
57	56	anim2d 613	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → ((𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)) → (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
58	57	reximdv 3153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → (∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
59	58	adantl 481	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . 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 684	. . . . . . . . . . . . 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 3140	. . . . . . . . . . . 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 3139	. . . . . . . . . . 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 412	. . . . . . . . . 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 29	. . . . . . . . 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 406	. . . . . . . 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 592	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → (𝑥 ∈ (𝑢 ∩ 𝑣) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
69	68	ralrimiv 3129	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → ∀𝑥 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))
70	33, 69	jca 511	. . . . 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 412	. . . 4 ⊢ (𝐵 ∈ TopBases → ((𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵)) → ((𝑢 ∩ 𝑣) ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))))
72		eltg2 22914	. . . 4 ⊢ (𝐵 ∈ TopBases → ((𝑢 ∩ 𝑣) ∈ (topGen‘𝐵) ↔ ((𝑢 ∩ 𝑣) ⊆ ∪ 𝐵 ∧ ∀𝑥 ∈ (𝑢 ∩ 𝑣)∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣)))))
73	71, 72	sylibrd 259	. . 3 ⊢ (𝐵 ∈ TopBases → ((𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵)) → (𝑢 ∩ 𝑣) ∈ (topGen‘𝐵)))
74	73	ralrimivv 3179	. 2 ⊢ (𝐵 ∈ TopBases → ∀𝑢 ∈ (topGen‘𝐵)∀𝑣 ∈ (topGen‘𝐵)(𝑢 ∩ 𝑣) ∈ (topGen‘𝐵))
75		fvex 6855	. . 3 ⊢ (topGen‘𝐵) ∈ V
76		istopg 22851	. . 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 584	1 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1540 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 Vcvv 3442 ∩ cin 3902 ⊆ wss 3903 ∪ cuni 4865 ‘cfv 6500 topGenctg 17369 Topctop 22849 TopBasesctb 22901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-iota 6456 df-fun 6502 df-fv 6508 df-topgen 17375 df-top 22850 df-bases 22902

This theorem is referenced by: tgclb 22926 tgtopon 22927 bastop 22937 elcls3 23039 resttop 23116 leordtval2 23168 tgcmp 23357 2ndctop 23403 2ndcsb 23405 2ndcsep 23415 txtop 23525 pttop 23538 xkotop 23544 alexsubALT 24007 retop 24717 onsuctop 36649 kelac2lem 43421

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