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Theorem tgcl 22911

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 4869	. . . . . . . 8 ⊢ (𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ⊆ ∪ (topGen‘𝐵))
2	1	adantl 481	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ 𝑢 ⊆ ∪ (topGen‘𝐵))
3		unitg 22909	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵)
4	3	adantr 480	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ (topGen‘𝐵) = ∪ 𝐵)
5	2, 4	sseqtrd 3968	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑢 ⊆ (topGen‘𝐵)) → ∪ 𝑢 ⊆ ∪ 𝐵)
6		eluni2 4865	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑢 ↔ ∃𝑡 ∈ 𝑢 𝑥 ∈ 𝑡)
7		ssel2 3926	. . . . . . . . . . . 12 ⊢ ((𝑢 ⊆ (topGen‘𝐵) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ (topGen‘𝐵))
8		eltg2b 22901	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ TopBases → (𝑡 ∈ (topGen‘𝐵) ↔ ∀𝑥 ∈ 𝑡 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝑡)))
9		rsp 3222	. . . . . . . . . . . . . . 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 4892	. . . . . . . . . . . . . 14 ⊢ (𝑡 ∈ 𝑢 → 𝑡 ⊆ ∪ 𝑢)
16		sstr2 3938	. . . . . . . . . . . . . 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 3149	. . . . . . . . . . 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 3136	. . . . . . . 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 3125	. . . . . 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 22900	. . . 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 1928	. 2 ⊢ (𝐵 ∈ TopBases → ∀𝑢(𝑢 ⊆ (topGen‘𝐵) → ∪ 𝑢 ∈ (topGen‘𝐵)))
30		inss1 4187	. . . . . . . 8 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑢
31		tg1 22906	. . . . . . . 8 ⊢ (𝑢 ∈ (topGen‘𝐵) → 𝑢 ⊆ ∪ 𝐵)
32	30, 31	sstrid 3943	. . . . . . 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 22900	. . . . . . . . . . . . 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 3222	. . . . . . . . . . . 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 22900	. . . . . . . . . . . . 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 3222	. . . . . . . . . . . 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 3915	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑢 ∩ 𝑣) ↔ (𝑥 ∈ 𝑢 ∧ 𝑥 ∈ 𝑣))
44		reeanv 3206	. . . . . . . . . 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 3915	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝑧 ∩ 𝑤) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤))
48	47	biimpri 228	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑤) → 𝑥 ∈ (𝑧 ∩ 𝑤))
49		ss2in 4195	. . . . . . . . . . . . . . . 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 22893	. . . . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . . . . . 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 3136	. . . . . . . . . . . 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 3135	. . . . . . . . . . 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 3125	. . . . . 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 22900	. . . 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 3175	. 2 ⊢ (𝐵 ∈ TopBases → ∀𝑢 ∈ (topGen‘𝐵)∀𝑣 ∈ (topGen‘𝐵)(𝑢 ∩ 𝑣) ∈ (topGen‘𝐵))
75		fvex 6845	. . 3 ⊢ (topGen‘𝐵) ∈ V
76		istopg 22837	. . 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 1539 = wceq 1541 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 Vcvv 3438 ∩ cin 3898 ⊆ wss 3899 ∪ cuni 4861 ‘cfv 6490 topGenctg 17355 Topctop 22835 TopBasesctb 22887

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-iota 6446 df-fun 6492 df-fv 6498 df-topgen 17361 df-top 22836 df-bases 22888

This theorem is referenced by: tgclb 22912 tgtopon 22913 bastop 22923 elcls3 23025 resttop 23102 leordtval2 23154 tgcmp 23343 2ndctop 23389 2ndcsb 23391 2ndcsep 23401 txtop 23511 pttop 23524 xkotop 23530 alexsubALT 23993 retop 24703 onsuctop 36576 kelac2lem 43248

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