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Theorem tgcl 22923

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 4895	. . . . . . . 8 ⊢ (𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ⊆ ∪ (topGen‘𝐵))
2	1	adantl 481	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ 𝑢 ⊆ ∪ (topGen‘𝐵))
3		unitg 22921	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵)
4	3	adantr 480	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ (topGen‘𝐵) = ∪ 𝐵)
5	2, 4	sseqtrd 4000	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ 𝑢 ⊆ ∪ 𝐵)
6		eluni2 4891	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑢 ↔ ∃𝑡 ∈ 𝑢 𝑥 ∈ 𝑡)
7		ssel2 3958	. . . . . . . . . . . 12 ⊢ ((𝑢 ⊆ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ (topGen‘𝐵))
8		eltg2b 22913	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ TopBases → (𝑡 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡)))
9		rsp 3233	. . . . . . . . . . . . . . 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 652	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ TopBases ∧ 𝑥 ∈ 𝑡) ∧ 𝑡 ∈ (topGen‘𝐵)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
13	7, 12	sylan2 593	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑥 ∈ 𝑡) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑢)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
14	13	an42s 661	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑥 ∈ 𝑡)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
15		elssuni 4917	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑢 → 𝑡 ⊆ ∪ 𝑢)
16		sstr2 3970	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝑡 → (𝑡 ⊆ ∪ 𝑢 → 𝑦 ⊆ ∪ 𝑢))
17	15, 16	syl5com 31	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑢 → (𝑦 ⊆ 𝑡 → 𝑦 ⊆ ∪ 𝑢))
18	17	anim2d 612	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝑢 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡) → (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
19	18	reximdv 3157	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝑢 → (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
20	19	ad2antrl 728	. . . . . . . . . 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 3143	. . . . . . . 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 3132	. . . . . 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 22912	. . . 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 1926	. 2 ⊢ (𝐵 ∈ TopBases → ∀𝑢(𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ∈ (topGen‘𝐵)))
30		inss1 4217	. . . . . . . 8 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑢
31		tg1 22918	. . . . . . . 8 ⊢ (𝑢 ∈ (topGen‘𝐵) → 𝑢 ⊆ ∪ 𝐵)
32	30, 31	sstrid 3975	. . . . . . 7 ⊢ (𝑢 ∈ (topGen‘𝐵) → (𝑢 ∩ 𝑣) ⊆ ∪ 𝐵)
33	32	ad2antrl 728	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → (𝑢 ∩ 𝑣) ⊆ ∪ 𝐵)
34		eltg2 22912	. . . . . . . . . . . . 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 3233	. . . . . . . . . . . 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 22912	. . . . . . . . . . . . 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 3233	. . . . . . . . . . . 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 620	. . . . . . . . . 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 3947	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑢 ∩ 𝑣) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))
44		reeanv 3216	. . . . . . . . . 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 678	. . . . . . . 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 3947	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑧 ∩ 𝑤) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤))
48	47	biimpri 228	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (𝑧 ∩ 𝑤))
49		ss2in 4225	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣) → (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣))
50	48, 49	anim12i 613	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤) ∧ (𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣)) → (𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)))
51	50	an4s 660	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)))
52		basis2 22905	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ TopBases ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ (𝑧 ∩ 𝑤))) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)))
53	52	adantllr 719	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ (𝑧 ∩ 𝑤))) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)))
54	53	adantrrr 725	. . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . . . . . . . . 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 612	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → ((𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)) → (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
58	57	reximdv 3157	. . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . 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 683	. . . . . . . . . . . . 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 3143	. . . . . . . . . . . 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 3142	. . . . . . . . . . 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 591	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → (𝑥 ∈ (𝑢 ∩ 𝑣) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
69	68	ralrimiv 3132	. . . . . 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 22912	. . . 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 3187	. 2 ⊢ (𝐵 ∈ TopBases → ∀𝑢 ∈ (topGen‘𝐵)∀𝑣 ∈ (topGen‘𝐵)(𝑢 ∩ 𝑣) ∈ (topGen‘𝐵))
75		fvex 6899	. . 3 ⊢ (topGen‘𝐵) ∈ V
76		istopg 22849	. . 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 583	1 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1537 = wceq 1539 ∈ wcel 2107 ∀wral 3050 ∃wrex 3059 Vcvv 3463 ∩ cin 3930 ⊆ wss 3931 ∪ cuni 4887 ‘cfv 6541 topGenctg 17453 Topctop 22847 TopBasesctb 22899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rab 3420 df-v 3465 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-br 5124 df-opab 5186 df-mpt 5206 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-iota 6494 df-fun 6543 df-fv 6549 df-topgen 17459 df-top 22848 df-bases 22900

This theorem is referenced by: tgclb 22924 tgtopon 22925 bastop 22935 elcls3 23037 resttop 23114 leordtval2 23166 tgcmp 23355 2ndctop 23401 2ndcsb 23403 2ndcsep 23413 txtop 23523 pttop 23536 xkotop 23542 alexsubALT 24005 retop 24718 onsuctop 36393 kelac2lem 43039

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