Theorem tgcmp 23724

Description: A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 24369, 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 2760	. . . . 5 ⊢ ∪ (topGen‘𝐵) = ∪ (topGen‘𝐵)
2	1	iscmp 23711	. . . 4 ⊢ ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑦 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧)))
3	2	simprbi 503	. . 3 ⊢ ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧))
4		unitg 23290	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵)
5		eqtr3 2782	. . . . . . . 8 ⊢ ((∪ (topGen‘𝐵) = ∪ 𝐵 ∧ 𝑋 = ∪ 𝐵) → ∪ (topGen‘𝐵) = 𝑋)
6	4, 5	sylan 592	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ∪ (topGen‘𝐵) = 𝑋)
7	6	eqeq1d 2762	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
8	6	eqeq1d 2762	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑧 ↔ 𝑋 = ∪ 𝑧))
9	8	rexbidv 3186	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
10	7, 9	imbi12d 347	. . . . 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 3185	. . . 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 23289	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
13	12	adantr 486	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (topGen‘𝐵))
14	13	sspwd 4569	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵))
15		ssralv 3977	. . . . 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 18	. . . 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 243	. . 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 35	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
19		elpwi 4563	. . . . 5 ⊢ (𝑢 ∈ 𝒫 (topGen‘𝐵) → 𝑢 ⊆ (topGen‘𝐵))
20		simprr 785	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ 𝑢)
21		simprl 783	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑢 ⊆ (topGen‘𝐵))
22	21	sselda 3908	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ (topGen‘𝐵))
23	22	adantrr 730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ∈ (topGen‘𝐵))
24		simprr 785	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡)
25		tg2 23288	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ 𝑡) → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
26	23, 24, 25	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
27	26	expr 462	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ 𝑡 ∈ 𝑢) → (𝑦 ∈ 𝑡 → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡)))
28	27	reximdva 3175	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → (∃𝑡 ∈ 𝑢 𝑦 ∈ 𝑡 → ∃𝑡 ∈ 𝑢 ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡)))
29		eluni2 4870	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ∪ 𝑢 ↔ ∃𝑡 ∈ 𝑢 𝑦 ∈ 𝑡)
30		elunirab 4881	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ↔ ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
31		r19.42v 3194	. . . . . . . . . . . . . . 15 ⊢ (∃𝑡 ∈ 𝑢 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡) ↔ (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
32	31	rexbii 3109	. . . . . . . . . . . . . 14 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑡 ∈ 𝑢 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡) ↔ ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
33		rexcom 3291	. . . . . . . . . . . . . 14 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑡 ∈ 𝑢 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡) ↔ ∃𝑡 ∈ 𝑢 ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
34	30, 32, 33	3bitr2i 302	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ↔ ∃𝑡 ∈ 𝑢 ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
35	28, 29, 34	3imtr4g 299	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → (𝑦 ∈ ∪ 𝑢 → 𝑦 ∈ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡}))
36	35	ssrdv 3914	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → ∪ 𝑢 ⊆ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
37	20, 36	eqsstrd 3942	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 ⊆ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
38		ssrab2 4005	. . . . . . . . . . . 12 ⊢ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝐵
39	38	unissi 4875	. . . . . . . . . . 11 ⊢ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ ∪ 𝐵
40		simplr 781	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ 𝐵)
41	39, 40	sseqtrrid 3951	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝑋)
42	37, 41	eqssd 3925	. . . . . . . . 9 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
43		elpw2g 5294	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ TopBases → ({𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∈ 𝒫 𝐵 ↔ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝐵))
44	43	ad2antrr 739	. . . . . . . . . . 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 261	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∈ 𝒫 𝐵)
46		unieq 4877	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → ∪ 𝑦 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
47	46	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡}))
48		pweq 4570	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → 𝒫 𝑦 = 𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
49	48	ineq1d 4116	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → (𝒫 𝑦 ∩ Fin) = (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin))
50	49	rexeqdv 3320	. . . . . . . . . . . 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 347	. . . . . . . . . . 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 3572	. . . . . . . . . 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 18	. . . . . . . . 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 45	. . . . . . . 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 9346	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ↔ (𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∧ 𝑧 ∈ Fin))
56	55	simprbi 503	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) → 𝑧 ∈ Fin)
57	56	ad2antrl 741	. . . . . . . . . . 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 502	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) → 𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
59	58	ad2antrl 741	. . . . . . . . . . . 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 3996	. . . . . . . . . . . . 13 ⊢ (𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ↔ (𝑧 ⊆ 𝐵 ∧ ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
61	60	simprbi 503	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡)
62	59, 61	syl 18	. . . . . . . . . . 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 3934	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑓‘𝑤) → (𝑤 ⊆ 𝑡 ↔ 𝑤 ⊆ (𝑓‘𝑤)))
64	63	ac6sfi 9278	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡) → ∃𝑓(𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤)))
65	57, 62, 64	syl2anc 596	. . . . . . . . . 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 6717	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑧⟶𝑢 → ran 𝑓 ⊆ 𝑢)
67	66	ad2antrl 741	. . . . . . . . . . . 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 6709	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑧⟶𝑢 → 𝑓 Fn 𝑧)
69		dffn4 6802	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fn 𝑧 ↔ 𝑓:𝑧–onto→ran 𝑓)
70	68, 69	sylib 221	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝑧⟶𝑢 → 𝑓:𝑧–onto→ran 𝑓)
71	70	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤)) → 𝑓:𝑧–onto→ran 𝑓)
72		fofi 9308	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Fin ∧ 𝑓:𝑧–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
73	57, 71, 72	syl2an 608	. . . . . . . . . . . 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 9346	. . . . . . . . . . . 12 ⊢ (ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran 𝑓 ⊆ 𝑢 ∧ ran 𝑓 ∈ Fin))
75	67, 73, 74	sylanbrc 595	. . . . . . . . . . 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 790	. . . . . . . . . . . . 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 5016	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 = ∪ 𝑤 ∈ 𝑧 𝑤
78		ss2iun 4969	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤) → ∪ 𝑤 ∈ 𝑧 𝑤 ⊆ ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤))
79	77, 78	eqsstrid 3946	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤) → ∪ 𝑧 ⊆ ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤))
80	79	ad2antll 742	. . . . . . . . . . . . . 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 7251	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn 𝑧 → ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤) = ∪ ran 𝑓)
82	68, 81	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑧⟶𝑢 → ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤) = ∪ ran 𝑓)
83	82	ad2antrl 741	. . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . 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 3942	. . . . . . . . . . . 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 4876	. . . . . . . . . . . . 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 739	. . . . . . . . . . . . 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 3945	. . . . . . . . . . . 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 3925	. . . . . . . . . . 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 4877	. . . . . . . . . . . 12 ⊢ (𝑣 = ran 𝑓 → ∪ 𝑣 = ∪ ran 𝑓)
91	90	rspceeqv 3598	. . . . . . . . . . 11 ⊢ ((ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑋 = ∪ ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)
92	75, 89, 91	syl2anc 596	. . . . . . . . . 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 1967	. . . . . . . . 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 3164	. . . . . . . 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 48	. . . . . . 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 462	. . . . . 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 87	. . . . 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 605	. . . 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 3162	. . 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 23292	. . . . . 6 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
101	100	adantr 486	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (topGen‘𝐵) ∈ Top)
102	1	iscmp 23711	. . . . . 6 ⊢ ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣)))
103	102	baib 545	. . . . 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 18	. . . 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 2762	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑢 ↔ 𝑋 = ∪ 𝑢))
106	6	eqeq1d 2762	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑣 ↔ 𝑋 = ∪ 𝑣))
107	106	rexbidv 3186	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣))
108	105, 107	imbi12d 347	. . . . 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 3185	. . . 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 282	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)))
111	99, 110	sylibrd 262	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → (topGen‘𝐵) ∈ Comp))
112	18, 111	impbid 215	1 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556  ∪ cuni 4866  ∪ ciun 4950  ran crn 5651   Fn wfn 6533  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538  Fincfn 8976  topGenctg 17613  Topctop 23216  TopBasesctb 23268  Compccmp 23709

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8479  df-en 8977  df-dom 8978  df-fin 8980  df-topgen 17619  df-top 23217  df-bases 23269  df-cmp 23710

This theorem is used by: (None)