Theorem tgcmp 21204

Description: A topology generated by a basis is compact iff open covers drawn from the basis have finite subcovers. (See also alexsub 21849, 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 2622	. . . . 5 ⊢ ∪ (topGen‘𝐵) = ∪ (topGen‘𝐵)
2	1	iscmp 21191	. . . 4 ⊢ ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑦 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧)))
3	2	simprbi 480	. . 3 ⊢ ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧))
4		unitg 20771	. . . . . . . 8 ⊢ (𝐵 ∈ TopBases → ∪ (topGen‘𝐵) = ∪ 𝐵)
5		eqtr3 2643	. . . . . . . 8 ⊢ ((∪ (topGen‘𝐵) = ∪ 𝐵 ∧ 𝑋 = ∪ 𝐵) → ∪ (topGen‘𝐵) = 𝑋)
6	4, 5	sylan 488	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ∪ (topGen‘𝐵) = 𝑋)
7	6	eqeq1d 2624	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑦))
8	6	eqeq1d 2624	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑧 ↔ 𝑋 = ∪ 𝑧))
9	8	rexbidv 3052	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))
10	7, 9	imbi12d 334	. . . . 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 2986	. . . 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 20770	. . . . . . 7 ⊢ (𝐵 ∈ TopBases → 𝐵 ⊆ (topGen‘𝐵))
13	12	adantr 481	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (topGen‘𝐵))
14		sspwb 4917	. . . . . 6 ⊢ (𝐵 ⊆ (topGen‘𝐵) ↔ 𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵))
15	13, 14	sylib 208	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘𝐵))
16		ssralv 3666	. . . . 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)))
17	15, 16	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)))
18	11, 17	sylbid 230	. . 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)))
19	3, 18	syl5 34	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp → ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))
20		elpwi 4168	. . . . 5 ⊢ (𝑢 ∈ 𝒫 (topGen‘𝐵) → 𝑢 ⊆ (topGen‘𝐵))
21		simprr 796	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ 𝑢)
22		simprl 794	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑢 ⊆ (topGen‘𝐵))
23	22	sselda 3603	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ (topGen‘𝐵))
24	23	adantrr 753	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → 𝑡 ∈ (topGen‘𝐵))
25		simprr 796	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → 𝑦 ∈ 𝑡)
26		tg2 20769	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡 ∈ (topGen‘𝐵) ∧ 𝑦 ∈ 𝑡) → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
27	24, 25, 26	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑡 ∈ 𝑢 ∧ 𝑦 ∈ 𝑡)) → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
28	27	expr 643	. . . . . . . . . . . . . 14 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ 𝑡 ∈ 𝑢) → (𝑦 ∈ 𝑡 → ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡)))
29	28	reximdva 3017	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → (∃𝑡 ∈ 𝑢 𝑦 ∈ 𝑡 → ∃𝑡 ∈ 𝑢 ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡)))
30		eluni2 4440	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ∪ 𝑢 ↔ ∃𝑡 ∈ 𝑢 𝑦 ∈ 𝑡)
31		elunirab 4448	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ↔ ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
32		r19.42v 3092	. . . . . . . . . . . . . . 15 ⊢ (∃𝑡 ∈ 𝑢 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡) ↔ (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
33	32	rexbii 3041	. . . . . . . . . . . . . 14 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑡 ∈ 𝑢 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡) ↔ ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
34		rexcom 3099	. . . . . . . . . . . . . 14 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑡 ∈ 𝑢 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡) ↔ ∃𝑡 ∈ 𝑢 ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
35	31, 33, 34	3bitr2i 288	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ↔ ∃𝑡 ∈ 𝑢 ∃𝑤 ∈ 𝐵 (𝑦 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑡))
36	29, 30, 35	3imtr4g 285	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → (𝑦 ∈ ∪ 𝑢 → 𝑦 ∈ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡}))
37	36	ssrdv 3609	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → ∪ 𝑢 ⊆ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
38	21, 37	eqsstrd 3639	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 ⊆ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
39		ssrab2 3687	. . . . . . . . . . . 12 ⊢ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝐵
40	39	unissi 4461	. . . . . . . . . . 11 ⊢ ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ ∪ 𝐵
41		simplr 792	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ 𝐵)
42	40, 41	syl5sseqr 3654	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝑋)
43	38, 42	eqssd 3620	. . . . . . . . 9 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → 𝑋 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
44		elpw2g 4827	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ TopBases → ({𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∈ 𝒫 𝐵 ↔ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ⊆ 𝐵))
45	44	ad2antrr 762	. . . . . . . . . . 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))
46	39, 45	mpbiri 248	. . . . . . . . . 10 ⊢ (((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) → {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∈ 𝒫 𝐵)
47		unieq 4444	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → ∪ 𝑦 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
48	47	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡}))
49		pweq 4161	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → 𝒫 𝑦 = 𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
50	49	ineq1d 3813	. . . . . . . . . . . . 13 ⊢ (𝑦 = {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → (𝒫 𝑦 ∩ Fin) = (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin))
51	50	rexeqdv 3145	. . . . . . . . . . . 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))
52	48, 51	imbi12d 334	. . . . . . . . . . 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)))
53	52	rspcv 3305	. . . . . . . . . 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)))
54	46, 53	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)))
55	43, 54	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))
56		elfpw 8268	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ↔ (𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∧ 𝑧 ∈ Fin))
57	56	simprbi 480	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) → 𝑧 ∈ Fin)
58	57	ad2antrl 764	. . . . . . . . . . 11 ⊢ ((((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) ∧ (𝑢 ⊆ (topGen‘𝐵) ∧ 𝑋 = ∪ 𝑢)) ∧ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) ∧ 𝑋 = ∪ 𝑧)) → 𝑧 ∈ Fin)
59	56	simplbi 476	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝒫 {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ∩ Fin) → 𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡})
60	59	ad2antrl 764	. . . . . . . . . . . 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})
61		ssrab 3680	. . . . . . . . . . . . 13 ⊢ (𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} ↔ (𝑧 ⊆ 𝐵 ∧ ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡))
62	61	simprbi 480	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ {𝑤 ∈ 𝐵 ∣ ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡} → ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡)
63	60, 62	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)
64		sseq2 3627	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑓‘𝑤) → (𝑤 ⊆ 𝑡 ↔ 𝑤 ⊆ (𝑓‘𝑤)))
65	64	ac6sfi 8204	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ Fin ∧ ∀𝑤 ∈ 𝑧 ∃𝑡 ∈ 𝑢 𝑤 ⊆ 𝑡) → ∃𝑓(𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤)))
66	58, 63, 65	syl2anc 693	. . . . . . . . . 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)))
67		frn 6053	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑧⟶𝑢 → ran 𝑓 ⊆ 𝑢)
68	67	ad2antrl 764	. . . . . . . . . . . 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)
69		ffn 6045	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑧⟶𝑢 → 𝑓 Fn 𝑧)
70		dffn4 6121	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fn 𝑧 ↔ 𝑓:𝑧–onto→ran 𝑓)
71	69, 70	sylib 208	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝑧⟶𝑢 → 𝑓:𝑧–onto→ran 𝑓)
72	71	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑧⟶𝑢 ∧ ∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤)) → 𝑓:𝑧–onto→ran 𝑓)
73		fofi 8252	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ Fin ∧ 𝑓:𝑧–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
74	58, 72, 73	syl2an 494	. . . . . . . . . . . 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)
75		elfpw 8268	. . . . . . . . . . . 12 ⊢ (ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran 𝑓 ⊆ 𝑢 ∧ ran 𝑓 ∈ Fin))
76	68, 74, 75	sylanbrc 698	. . . . . . . . . . 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))
77		simplrr 801	. . . . . . . . . . . . 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)
78		uniiun 4573	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑧 = ∪ 𝑤 ∈ 𝑧 𝑤
79		ss2iun 4536	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤) → ∪ 𝑤 ∈ 𝑧 𝑤 ⊆ ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤))
80	78, 79	syl5eqss 3649	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ 𝑧 𝑤 ⊆ (𝑓‘𝑤) → ∪ 𝑧 ⊆ ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤))
81	80	ad2antll 765	. . . . . . . . . . . . . 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))
82		fniunfv 6505	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn 𝑧 → ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤) = ∪ ran 𝑓)
83	69, 82	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑧⟶𝑢 → ∪ 𝑤 ∈ 𝑧 (𝑓‘𝑤) = ∪ ran 𝑓)
84	83	ad2antrl 764	. . . . . . . . . . . . . 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)
85	81, 84	sseqtrd 3641	. . . . . . . . . . . . 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)
86	77, 85	eqsstrd 3639	. . . . . . . . . . . 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)
87	68	unissd 4462	. . . . . . . . . . . . 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)
88	21	ad2antrr 762	. . . . . . . . . . . . 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)
89	87, 88	sseqtr4d 3642	. . . . . . . . . . . 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)
90	86, 89	eqssd 3620	. . . . . . . . . . 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)
91		unieq 4444	. . . . . . . . . . . . 13 ⊢ (𝑣 = ran 𝑓 → ∪ 𝑣 = ∪ ran 𝑓)
92	91	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑣 = ran 𝑓 → (𝑋 = ∪ 𝑣 ↔ 𝑋 = ∪ ran 𝑓))
93	92	rspcev 3309	. . . . . . . . . . 11 ⊢ ((ran 𝑓 ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑋 = ∪ ran 𝑓) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)
94	76, 90, 93	syl2anc 693	. . . . . . . . . 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)
95	66, 94	exlimddv 1863	. . . . . . . . 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)
96	95	rexlimdvaa 3032	. . . . . . . 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))
97	55, 96	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))
98	97	expr 643	. . . . . 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)))
99	98	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)))
100	20, 99	sylan2 491	. . . 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)))
101	100	ralrimdva 2969	. . 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)))
102		tgcl 20773	. . . . . 6 ⊢ (𝐵 ∈ TopBases → (topGen‘𝐵) ∈ Top)
103	102	adantr 481	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (topGen‘𝐵) ∈ Top)
104	1	iscmp 21191	. . . . . 6 ⊢ ((topGen‘𝐵) ∈ Comp ↔ ((topGen‘𝐵) ∈ Top ∧ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣)))
105	104	baib 944	. . . . 5 ⊢ ((topGen‘𝐵) ∈ Top → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣)))
106	103, 105	syl 17	. . . 4 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(∪ (topGen‘𝐵) = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣)))
107	6	eqeq1d 2624	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑢 ↔ 𝑋 = ∪ 𝑢))
108	6	eqeq1d 2624	. . . . . . 7 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∪ (topGen‘𝐵) = ∪ 𝑣 ↔ 𝑋 = ∪ 𝑣))
109	108	rexbidv 3052	. . . . . 6 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣))
110	107, 109	imbi12d 334	. . . . 5 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((∪ (topGen‘𝐵) = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)∪ (topGen‘𝐵) = ∪ 𝑣) ↔ (𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)))
111	110	ralbidv 2986	. . . 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)))
112	106, 111	bitrd 268	. . 3 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘𝐵)(𝑋 = ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑋 = ∪ 𝑣)))
113	101, 112	sylibrd 249	. 2 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → (∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧) → (topGen‘𝐵) ∈ Comp))
114	19, 113	impbid 202	1 ⊢ ((𝐵 ∈ TopBases ∧ 𝑋 = ∪ 𝐵) → ((topGen‘𝐵) ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ∪ ciun 4520  ran crn 5115   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  Fincfn 7955  topGenctg 16098  Topctop 20698  TopBasesctb 20749  Compccmp 21189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959  df-topgen 16104  df-top 20699  df-bases 20750  df-cmp 21190

This theorem is referenced by: (None)