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Theorem tgcl 22997

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 4939	. . . . . . . 8 ⊢ (𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ⊆ ∪ (topGen‘𝐵))
2	1	adantl 481	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ 𝑢 ⊆ ∪ (topGen‘𝐵))
3		unitg 22995	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵)
4	3	adantr 480	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ (topGen‘𝐵) = ∪ 𝐵)
5	2, 4	sseqtrd 4049	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ 𝑢 ⊆ ∪ 𝐵)
6		eluni2 4935	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑢 ↔ ∃𝑡 ∈ 𝑢 𝑥 ∈ 𝑡)
7		ssel2 4003	. . . . . . . . . . . 12 ⊢ ((𝑢 ⊆ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ (topGen‘𝐵))
8		eltg2b 22987	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ TopBases → (𝑡 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡)))
9		rsp 3253	. . . . . . . . . . . . . . 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 651	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ TopBases ∧ 𝑥 ∈ 𝑡) ∧ 𝑡 ∈ (topGen‘𝐵)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
13	7, 12	sylan2 592	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑥 ∈ 𝑡) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑢)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
14	13	an42s 660	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑥 ∈ 𝑡)) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡))
15		elssuni 4961	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑢 → 𝑡 ⊆ ∪ 𝑢)
16		sstr2 4015	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 𝑡 → (𝑡 ⊆ ∪ 𝑢 → 𝑦 ⊆ ∪ 𝑢))
17	15, 16	syl5com 31	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝑢 → (𝑦 ⊆ 𝑡 → 𝑦 ⊆ ∪ 𝑢))
18	17	anim2d 611	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝑢 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡) → (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
19	18	reximdv 3176	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝑢 → (∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡) → ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ ∪ 𝑢)))
20	19	ad2antrl 727	. . . . . . . . . 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 3162	. . . . . . . 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 3151	. . . . . 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 22986	. . . 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 4258	. . . . . . . 8 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑢
31		tg1 22992	. . . . . . . 8 ⊢ (𝑢 ∈ (topGen‘𝐵) → 𝑢 ⊆ ∪ 𝐵)
32	30, 31	sstrid 4020	. . . . . . 7 ⊢ (𝑢 ∈ (topGen‘𝐵) → (𝑢 ∩ 𝑣) ⊆ ∪ 𝐵)
33	32	ad2antrl 727	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → (𝑢 ∩ 𝑣) ⊆ ∪ 𝐵)
34		eltg2 22986	. . . . . . . . . . . . 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 3253	. . . . . . . . . . . 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 22986	. . . . . . . . . . . . 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 3253	. . . . . . . . . . . 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 619	. . . . . . . . . 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 3992	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑢 ∩ 𝑣) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))
44		reeanv 3235	. . . . . . . . . 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 677	. . . . . . . 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 3992	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑧 ∩ 𝑤) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤))
48	47	biimpri 228	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (𝑧 ∩ 𝑤))
49		ss2in 4266	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣) → (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣))
50	48, 49	anim12i 612	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤) ∧ (𝑧 ⊆ 𝑢 ∧ 𝑤 ⊆ 𝑣)) → (𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)))
51	50	an4s 659	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ∧ (𝑥 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑣)) → (𝑥 ∈ (𝑧 ∩ 𝑤) ∧ (𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣)))
52		basis2 22979	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ TopBases ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ (𝑧 ∩ 𝑤))) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)))
53	52	adantllr 718	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑥 ∈ (𝑢 ∩ 𝑣)) ∧ 𝑧 ∈ 𝐵) ∧ (𝑤 ∈ 𝐵 ∧ 𝑥 ∈ (𝑧 ∩ 𝑤))) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)))
54	53	adantrrr 724	. . . . . . . . . . . . . . 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 4015	. . . . . . . . . . . . . . . . . . . 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 611	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∩ 𝑤) ⊆ (𝑢 ∩ 𝑣) → ((𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑧 ∩ 𝑤)) → (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
58	57	reximdv 3176	. . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . 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 682	. . . . . . . . . . . . 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 3162	. . . . . . . . . . . 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 3161	. . . . . . . . . . 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 590	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ (𝑢 ∈ (topGen‘𝐵) ∧ 𝑣 ∈ (topGen‘𝐵))) → (𝑥 ∈ (𝑢 ∩ 𝑣) → ∃𝑡 ∈ 𝐵 (𝑥 ∈ 𝑡 ∧ 𝑡 ⊆ (𝑢 ∩ 𝑣))))
69	68	ralrimiv 3151	. . . . . 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 22986	. . . 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 3206	. 2 ⊢ (𝐵 ∈ TopBases → ∀𝑢 ∈ (topGen‘𝐵)∀𝑣 ∈ (topGen‘𝐵)(𝑢 ∩ 𝑣) ∈ (topGen‘𝐵))
75		fvex 6933	. . 3 ⊢ (topGen‘𝐵) ∈ V
76		istopg 22922	. . 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 582	1 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ∩ cin 3975 ⊆ wss 3976 ∪ cuni 4931 ‘cfv 6573 topGenctg 17497 Topctop 22920 TopBasesctb 22973

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

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 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-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-iota 6525 df-fun 6575 df-fv 6581 df-topgen 17503 df-top 22921 df-bases 22974

This theorem is referenced by: tgclb 22998 tgtopon 22999 bastop 23009 elcls3 23112 resttop 23189 leordtval2 23241 tgcmp 23430 2ndctop 23476 2ndcsb 23478 2ndcsep 23488 txtop 23598 pttop 23611 xkotop 23617 alexsubALT 24080 retop 24803 onsuctop 36399 kelac2lem 43021

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