Theorem tgcmp 22145

Description: A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 22789, 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 2738	. . . . 5 ⊢ ∪ (topGen‘𝐵) = ∪ (topGen‘𝐵)
2	1	iscmp 22132	. . . 4 ⊢ ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑦 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧)))
3	2	simprbi 500	. . 3 ⊢ ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧))
4		unitg 21711	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵)
5		eqtr3 2760	. . . . . . . 8 ⊢ ((∪ (topGen‘𝐵) = ∪ 𝐵 ∧ 𝑋 = ∪ 𝐵) → ∪ (topGen‘𝐵) = 𝑋)
6	4, 5	sylan 583	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ∪ (topGen‘𝐵) = 𝑋)
7	6	eqeq1d 2740	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
8	6	eqeq1d 2740	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑧 ↔ 𝑋 = ∪ 𝑧))
9	8	rexbidv 3206	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
10	7, 9	imbi12d 348	. . . . 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 3109	. . . 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 21710	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
13	12	adantr 484	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (topGen‘𝐵))
14	13	sspwd 4500	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵))
15		ssralv 3941	. . . . 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 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 34	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
19		elpwi 4494	. . . . 5 ⊢ (𝑢 ∈ 𝒫 (topGen‘𝐵) → 𝑢 ⊆ (topGen‘𝐵))
20		simprr 773	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ 𝑢)
21		simprl 771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑢 ⊆ (topGen‘𝐵))
22	21	sselda 3875	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ (topGen‘𝐵))
23	22	adantrr 717	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ∈ (topGen‘𝐵))
24		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡)
25		tg2 21709	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ 𝑡) → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
26	23, 24, 25	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
27	26	expr 460	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ 𝑡 ∈ 𝑢) → (𝑦 ∈ 𝑡 → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡)))
28	27	reximdva 3183	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → (∃𝑡 ∈ 𝑢 𝑦 ∈ 𝑡 → ∃𝑡 ∈ 𝑢 ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡)))
29		eluni2 4797	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ∪ 𝑢 ↔ ∃𝑡 ∈ 𝑢 𝑦 ∈ 𝑡)
30		elunirab 4809	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ↔ ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
31		r19.42v 3253	. . . . . . . . . . . . . . 15 ⊢ (∃𝑡 ∈ 𝑢 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡) ↔ (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
32	31	rexbii 3160	. . . . . . . . . . . . . 14 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑡 ∈ 𝑢 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡) ↔ ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
33		rexcom 3258	. . . . . . . . . . . . . 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 3881	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → ∪ 𝑢 ⊆ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
37	20, 36	eqsstrd 3913	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 ⊆ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
38		ssrab2 3967	. . . . . . . . . . . 12 ⊢ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝐵
39	38	unissi 4802	. . . . . . . . . . 11 ⊢ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ ∪ 𝐵
40		simplr 769	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ 𝐵)
41	39, 40	sseqtrrid 3928	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝑋)
42	37, 41	eqssd 3892	. . . . . . . . 9 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
43		elpw2g 5209	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ TopBases → ({𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∈ 𝒫 𝐵 ↔ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝐵))
44	43	ad2antrr 726	. . . . . . . . . . 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 4804	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → ∪ 𝑦 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
47	46	eqeq2d 2749	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡}))
48		pweq 4501	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → 𝒫 𝑦 = 𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
49	48	ineq1d 4100	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → (𝒫 𝑦 ∩ Fin) = (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin))
50	49	rexeqdv 3316	. . . . . . . . . . . 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 348	. . . . . . . . . . 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 3519	. . . . . . . . . 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 8892	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ↔ (𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∧ 𝑧 ∈ Fin))
56	55	simprbi 500	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) → 𝑧 ∈ Fin)
57	56	ad2antrl 728	. . . . . . . . . . 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 501	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) → 𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
59	58	ad2antrl 728	. . . . . . . . . . . 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 3960	. . . . . . . . . . . . 13 ⊢ (𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ↔ (𝑧 ⊆ 𝐵 ∧ ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
61	60	simprbi 500	. . . . . . . . . . . 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 3901	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑓‘𝑤) → (𝑤 ⊆ 𝑡 ↔ 𝑤 ⊆ (𝑓‘𝑤)))
64	63	ac6sfi 8829	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡) → ∃𝑓(𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤)))
65	57, 62, 64	syl2anc 587	. . . . . . . . . 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 6505	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑧⟶𝑢 → ran 𝑓 ⊆ 𝑢)
67	66	ad2antrl 728	. . . . . . . . . . . 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 6498	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑧⟶𝑢 → 𝑓 Fn 𝑧)
69		dffn4 6592	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤)) → 𝑓:𝑧–onto→ran 𝑓)
72		fofi 8876	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Fin ∧ 𝑓:𝑧–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
73	57, 71, 72	syl2an 599	. . . . . . . . . . . 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 8892	. . . . . . . . . . . 12 ⊢ (ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran 𝑓 ⊆ 𝑢 ∧ ran 𝑓 ∈ Fin))
75	67, 73, 74	sylanbrc 586	. . . . . . . . . . 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 778	. . . . . . . . . . . . 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 4941	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 = ∪ 𝑤 ∈ 𝑧 𝑤
78		ss2iun 4896	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤) → ∪ 𝑤 ∈ 𝑧 𝑤 ⊆ ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤))
79	77, 78	eqsstrid 3923	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤) → ∪ 𝑧 ⊆ ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤))
80	79	ad2antll 729	. . . . . . . . . . . . . 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 7011	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . 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 3915	. . . . . . . . . . . . 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 3913	. . . . . . . . . . . 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 4803	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 3916	. . . . . . . . . . . 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 3892	. . . . . . . . . . 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 4804	. . . . . . . . . . . 12 ⊢ (𝑣 = ran 𝑓 → ∪ 𝑣 = ∪ ran 𝑓)
91	90	rspceeqv 3539	. . . . . . . . . . 11 ⊢ ((ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑋 = ∪ ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)
92	75, 89, 91	syl2anc 587	. . . . . . . . . 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 1941	. . . . . . . . 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 3194	. . . . . . . 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 460	. . . . . 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 596	. . . 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 3101	. . 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 21713	. . . . . 6 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
101	100	adantr 484	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (topGen‘𝐵) ∈ Top)
102	1	iscmp 22132	. . . . . 6 ⊢ ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣)))
103	102	baib 539	. . . . 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 2740	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑢 ↔ 𝑋 = ∪ 𝑢))
106	6	eqeq1d 2740	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑣 ↔ 𝑋 = ∪ 𝑣))
107	106	rexbidv 3206	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣))
108	105, 107	imbi12d 348	. . . . 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 3109	. . . 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)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542  ∃wex 1786   ∈ wcel 2113  ∀wral 3053  ∃wrex 3054  {crab 3057   ∩ cin 3840   ⊆ wss 3841  𝒫 cpw 4485  ∪ cuni 4793  ∪ ciun 4878  ran crn 5520   Fn wfn 6328  ⟶wf 6329  –onto→wfo 6331  ‘cfv 6333  Fincfn 8548  topGenctg 16807  Topctop 21637  TopBasesctb 21689  Compccmp 22130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-om 7594  df-1o 8124  df-er 8313  df-en 8549  df-dom 8550  df-fin 8552  df-topgen 16813  df-top 21638  df-bases 21690  df-cmp 22131

This theorem is referenced by: (None)