Theorem tgcmp 22896

Description: A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 23540, which further specializes to subbases, assuming the ultrafilter lemma.) (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
tgcmp	⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Distinct variable groups:   𝑦,𝑧,𝐵   𝑦,𝑋,𝑧

Proof of Theorem tgcmp

Dummy variables 𝑡 𝑓 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2732	. . . . 5 ⊢ ∪ (topGen‘𝐵) = ∪ (topGen‘𝐵)
2	1	iscmp 22883	. . . 4 ⊢ ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑦 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧)))
3	2	simprbi 497	. . 3 ⊢ ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧))
4		unitg 22461	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵)
5		eqtr3 2758	. . . . . . . 8 ⊢ ((∪ (topGen‘𝐵) = ∪ 𝐵 ∧ 𝑋 = ∪ 𝐵) → ∪ (topGen‘𝐵) = 𝑋)
6	4, 5	sylan 580	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ∪ (topGen‘𝐵) = 𝑋)
7	6	eqeq1d 2734	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
8	6	eqeq1d 2734	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑧 ↔ 𝑋 = ∪ 𝑧))
9	8	rexbidv 3178	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
10	7, 9	imbi12d 344	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((∪ (topGen‘𝐵) = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧) ↔ (𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
11	10	ralbidv 3177	. . . 4 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∀𝑦 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧) ↔ ∀𝑦 ∈ 𝒫 (topGen‘𝐵)(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
12		bastg 22460	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
13	12	adantr 481	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (topGen‘𝐵))
14	13	sspwd 4614	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵))
15		ssralv 4049	. . . . 5 ⊢ (𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵) → (∀𝑦 ∈ 𝒫 (topGen‘𝐵)(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
16	14, 15	syl 17	. . . 4 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∀𝑦 ∈ 𝒫 (topGen‘𝐵)(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
17	11, 16	sylbid 239	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∀𝑦 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧) → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
18	3, 17	syl5 34	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
19		elpwi 4608	. . . . 5 ⊢ (𝑢 ∈ 𝒫 (topGen‘𝐵) → 𝑢 ⊆ (topGen‘𝐵))
20		simprr 771	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ 𝑢)
21		simprl 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑢 ⊆ (topGen‘𝐵))
22	21	sselda 3981	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ (topGen‘𝐵))
23	22	adantrr 715	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ∈ (topGen‘𝐵))
24		simprr 771	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡)
25		tg2 22459	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ 𝑡) → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
26	23, 24, 25	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
27	26	expr 457	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ 𝑡 ∈ 𝑢) → (𝑦 ∈ 𝑡 → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡)))
28	27	reximdva 3168	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → (∃𝑡 ∈ 𝑢 𝑦 ∈ 𝑡 → ∃𝑡 ∈ 𝑢 ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡)))
29		eluni2 4911	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ∪ 𝑢 ↔ ∃𝑡 ∈ 𝑢 𝑦 ∈ 𝑡)
30		elunirab 4923	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ↔ ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
31		r19.42v 3190	. . . . . . . . . . . . . . 15 ⊢ (∃𝑡 ∈ 𝑢 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡) ↔ (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
32	31	rexbii 3094	. . . . . . . . . . . . . 14 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑡 ∈ 𝑢 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡) ↔ ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
33		rexcom 3287	. . . . . . . . . . . . . 14 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑡 ∈ 𝑢 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡) ↔ ∃𝑡 ∈ 𝑢 ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
34	30, 32, 33	3bitr2i 298	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ↔ ∃𝑡 ∈ 𝑢 ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
35	28, 29, 34	3imtr4g 295	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → (𝑦 ∈ ∪ 𝑢 → 𝑦 ∈ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡}))
36	35	ssrdv 3987	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → ∪ 𝑢 ⊆ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
37	20, 36	eqsstrd 4019	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 ⊆ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
38		ssrab2 4076	. . . . . . . . . . . 12 ⊢ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝐵
39	38	unissi 4916	. . . . . . . . . . 11 ⊢ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ ∪ 𝐵
40		simplr 767	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ 𝐵)
41	39, 40	sseqtrrid 4034	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝑋)
42	37, 41	eqssd 3998	. . . . . . . . 9 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
43		elpw2g 5343	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ TopBases → ({𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∈ 𝒫 𝐵 ↔ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝐵))
44	43	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → ({𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∈ 𝒫 𝐵 ↔ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝐵))
45	38, 44	mpbiri 257	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∈ 𝒫 𝐵)
46		unieq 4918	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → ∪ 𝑦 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
47	46	eqeq2d 2743	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡}))
48		pweq 4615	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → 𝒫 𝑦 = 𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
49	48	ineq1d 4210	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → (𝒫 𝑦 ∩ Fin) = (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin))
50	49	rexeqdv 3326	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin)𝑋 = ∪ 𝑧))
51	47, 50	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → ((𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) ↔ (𝑋 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → ∃𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin)𝑋 = ∪ 𝑧)))
52	51	rspcv 3608	. . . . . . . . . 10 ⊢ ({𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∈ 𝒫 𝐵 → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → (𝑋 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → ∃𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin)𝑋 = ∪ 𝑧)))
53	45, 52	syl 17	. . . . . . . . 9 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → (𝑋 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → ∃𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin)𝑋 = ∪ 𝑧)))
54	42, 53	mpid 44	. . . . . . . 8 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → ∃𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin)𝑋 = ∪ 𝑧))
55		elfpw 9350	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ↔ (𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∧ 𝑧 ∈ Fin))
56	55	simprbi 497	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) → 𝑧 ∈ Fin)
57	56	ad2antrl 726	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) → 𝑧 ∈ Fin)
58	55	simplbi 498	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) → 𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
59	58	ad2antrl 726	. . . . . . . . . . . 12 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) → 𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
60		ssrab 4069	. . . . . . . . . . . . 13 ⊢ (𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ↔ (𝑧 ⊆ 𝐵 ∧ ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
61	60	simprbi 497	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡)
62	59, 61	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) → ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡)
63		sseq2 4007	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑓‘𝑤) → (𝑤 ⊆ 𝑡 ↔ 𝑤 ⊆ (𝑓‘𝑤)))
64	63	ac6sfi 9283	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡) → ∃𝑓(𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤)))
65	57, 62, 64	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) → ∃𝑓(𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤)))
66		frn 6721	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑧⟶𝑢 → ran 𝑓 ⊆ 𝑢)
67	66	ad2antrl 726	. . . . . . . . . . . 12 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → ran 𝑓 ⊆ 𝑢)
68		ffn 6714	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑧⟶𝑢 → 𝑓 Fn 𝑧)
69		dffn4 6808	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fn 𝑧 ↔ 𝑓:𝑧–onto→ran 𝑓)
70	68, 69	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝑧⟶𝑢 → 𝑓:𝑧–onto→ran 𝑓)
71	70	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤)) → 𝑓:𝑧–onto→ran 𝑓)
72		fofi 9334	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Fin ∧ 𝑓:𝑧–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
73	57, 71, 72	syl2an 596	. . . . . . . . . . . 12 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → ran 𝑓 ∈ Fin)
74		elfpw 9350	. . . . . . . . . . . 12 ⊢ (ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran 𝑓 ⊆ 𝑢 ∧ ran 𝑓 ∈ Fin))
75	67, 73, 74	sylanbrc 583	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin))
76		simplrr 776	. . . . . . . . . . . . 13 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → 𝑋 = ∪ 𝑧)
77		uniiun 5060	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 = ∪ 𝑤 ∈ 𝑧 𝑤
78		ss2iun 5014	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤) → ∪ 𝑤 ∈ 𝑧 𝑤 ⊆ ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤))
79	77, 78	eqsstrid 4029	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤) → ∪ 𝑧 ⊆ ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤))
80	79	ad2antll 727	. . . . . . . . . . . . . 14 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → ∪ 𝑧 ⊆ ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤))
81		fniunfv 7242	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn 𝑧 → ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤) = ∪ ran 𝑓)
82	68, 81	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑧⟶𝑢 → ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤) = ∪ ran 𝑓)
83	82	ad2antrl 726	. . . . . . . . . . . . . 14 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤) = ∪ ran 𝑓)
84	80, 83	sseqtrd 4021	. . . . . . . . . . . . 13 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → ∪ 𝑧 ⊆ ∪ ran 𝑓)
85	76, 84	eqsstrd 4019	. . . . . . . . . . . 12 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → 𝑋 ⊆ ∪ ran 𝑓)
86	67	unissd 4917	. . . . . . . . . . . . 13 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → ∪ ran 𝑓 ⊆ ∪ 𝑢)
87	20	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → 𝑋 = ∪ 𝑢)
88	86, 87	sseqtrrd 4022	. . . . . . . . . . . 12 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → ∪ ran 𝑓 ⊆ 𝑋)
89	85, 88	eqssd 3998	. . . . . . . . . . 11 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → 𝑋 = ∪ ran 𝑓)
90		unieq 4918	. . . . . . . . . . . 12 ⊢ (𝑣 = ran 𝑓 → ∪ 𝑣 = ∪ ran 𝑓)
91	90	rspceeqv 3632	. . . . . . . . . . 11 ⊢ ((ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑋 = ∪ ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)
92	75, 89, 91	syl2anc 584	. . . . . . . . . 10 ⊢ (((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) ∧ (𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)
93	65, 92	exlimddv 1938	. . . . . . . . 9 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)
94	93	rexlimdvaa 3156	. . . . . . . 8 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → (∃𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin)𝑋 = ∪ 𝑧 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣))
95	54, 94	syld 47	. . . . . . 7 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣))
96	95	expr 457	. . . . . 6 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ 𝑢 ⊆ (topGen‘𝐵)) → (𝑋 = ∪ 𝑢 → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)))
97	96	com23 86	. . . . 5 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ 𝑢 ⊆ (topGen‘𝐵)) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → (𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)))
98	19, 97	sylan2 593	. . . 4 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ 𝑢 ∈ 𝒫 (topGen‘𝐵)) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → (𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)))
99	98	ralrimdva 3154	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)))
100		tgcl 22463	. . . . . 6 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
101	100	adantr 481	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (topGen‘𝐵) ∈ Top)
102	1	iscmp 22883	. . . . . 6 ⊢ ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣)))
103	102	baib 536	. . . . 5 ⊢ ((topGen‘𝐵) ∈ Top → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣)))
104	101, 103	syl 17	. . . 4 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣)))
105	6	eqeq1d 2734	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑢 ↔ 𝑋 = ∪ 𝑢))
106	6	eqeq1d 2734	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑣 ↔ 𝑋 = ∪ 𝑣))
107	106	rexbidv 3178	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣))
108	105, 107	imbi12d 344	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((∪ (topGen‘𝐵) = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣) ↔ (𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)))
109	108	ralbidv 3177	. . . 4 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∀𝑢 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣) ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)))
110	104, 109	bitrd 278	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)))
111	99, 110	sylibrd 258	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → (topGen‘𝐵) ∈ Comp))
112	18, 111	impbid 211	1 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  {crab 3432   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4601  ∪ cuni 4907  ∪ ciun 4996  ran crn 5676   Fn wfn 6535  ⟶wf 6536  –onto→wfo 6538  ‘cfv 6540  Fincfn 8935  topGenctg 17379  Topctop 22386  TopBasesctb 22439  Compccmp 22881

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-fin 8939  df-topgen 17385  df-top 22387  df-bases 22440  df-cmp 22882

This theorem is referenced by: (None)