Theorem iscldtop 23304

Description: A family is the closed sets of a topology iff it is a Moore collection and closed under finite union. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
iscldtop	⊢ (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) ↔ (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐾,𝑦

Proof of Theorem iscldtop

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fncld 23231	. . . . 5 ⊢ Clsd Fn Top
2		fnfun 6639	. . . . 5 ⊢ (Clsd Fn Top → Fun Clsd)
3	1, 2	ax-mp 5	. . . 4 ⊢ Fun Clsd
4		fvelima 6950	. . . 4 ⊢ ((Fun Clsd ∧ 𝐾 ∈ (Clsd “ (TopOn‘𝐵))) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
5	3, 4	mpan 703	. . 3 ⊢ (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
6		cldmreon 23303	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → (Clsd‘𝑎) ∈ (Moore‘𝐵))
7		topontop 23122	. . . . . . 7 ⊢ (𝑎 ∈ (TopOn‘𝐵) → 𝑎 ∈ Top)
8		0cld 23247	. . . . . . 7 ⊢ (𝑎 ∈ Top → ∅ ∈ (Clsd‘𝑎))
9	7, 8	syl 18	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ∅ ∈ (Clsd‘𝑎))
10		uncld 23250	. . . . . . . 8 ⊢ ((𝑥 ∈ (Clsd‘𝑎) ∧ 𝑦 ∈ (Clsd‘𝑎)) → (𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎))
11	10	adantl 487	. . . . . . 7 ⊢ ((𝑎 ∈ (TopOn‘𝐵) ∧ (𝑥 ∈ (Clsd‘𝑎) ∧ 𝑦 ∈ (Clsd‘𝑎))) → (𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎))
12	11	ralrimivva 3210	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎))
13	6, 9, 12	3jca 1146	. . . . 5 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ∧ ∅ ∈ (Clsd‘𝑎) ∧ ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎)))
14		eleq1 2853	. . . . . 6 ⊢ ((Clsd‘𝑎) = 𝐾 → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ↔ 𝐾 ∈ (Moore‘𝐵)))
15		eleq2 2854	. . . . . 6 ⊢ ((Clsd‘𝑎) = 𝐾 → (∅ ∈ (Clsd‘𝑎) ↔ ∅ ∈ 𝐾))
16		eleq2 2854	. . . . . . . 8 ⊢ ((Clsd‘𝑎) = 𝐾 → ((𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎) ↔ (𝑥 ∪ 𝑦) ∈ 𝐾))
17	16	raleqbi1dv 3335	. . . . . . 7 ⊢ ((Clsd‘𝑎) = 𝐾 → (∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))
18	17	raleqbi1dv 3335	. . . . . 6 ⊢ ((Clsd‘𝑎) = 𝐾 → (∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))
19	14, 15, 18	3anbi123d 1464	. . . . 5 ⊢ ((Clsd‘𝑎) = 𝐾 → (((Clsd‘𝑎) ∈ (Moore‘𝐵) ∧ ∅ ∈ (Clsd‘𝑎) ∧ ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎)) ↔ (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾)))
20	13, 19	syl5ibcom 248	. . . 4 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) = 𝐾 → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾)))
21	20	rexlimiv 3161	. . 3 ⊢ (∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾 → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))
22	5, 21	syl 18	. 2 ⊢ (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))
23		simp1 1154	. . . . 5 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → 𝐾 ∈ (Moore‘𝐵))
24		simp2 1155	. . . . 5 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → ∅ ∈ 𝐾)
25		uneq1 4115	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (𝑥 ∪ 𝑦) = (𝑏 ∪ 𝑦))
26	25	eleq1d 2850	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → ((𝑥 ∪ 𝑦) ∈ 𝐾 ↔ (𝑏 ∪ 𝑦) ∈ 𝐾))
27		uneq2 4116	. . . . . . . . . 10 ⊢ (𝑦 = 𝑐 → (𝑏 ∪ 𝑦) = (𝑏 ∪ 𝑐))
28	27	eleq1d 2850	. . . . . . . . 9 ⊢ (𝑦 = 𝑐 → ((𝑏 ∪ 𝑦) ∈ 𝐾 ↔ (𝑏 ∪ 𝑐) ∈ 𝐾))
29	26, 28	rspc2v 3594	. . . . . . . 8 ⊢ ((𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾 → (𝑏 ∪ 𝑐) ∈ 𝐾))
30	29	com12 33	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾 → ((𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (𝑏 ∪ 𝑐) ∈ 𝐾))
31	30	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → ((𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (𝑏 ∪ 𝑐) ∈ 𝐾))
32	31	3impib 1134	. . . . 5 ⊢ (((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (𝑏 ∪ 𝑐) ∈ 𝐾)
33		eqid 2765	. . . . 5 ⊢ {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾} = {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾}
34	23, 24, 32, 33	mretopd 23301	. . . 4 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) ∧ 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾})))
35	34	simprd 501	. . 3 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾}))
36	34	simpld 500	. . . 4 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵))
37	7	ssriv 3942	. . . . . 6 ⊢ (TopOn‘𝐵) ⊆ Top
38	1	fndmi 6643	. . . . . 6 ⊢ dom Clsd = Top
39	37, 38	sseqtrri 3987	. . . . 5 ⊢ (TopOn‘𝐵) ⊆ dom Clsd
40		funfvima2 7236	. . . . 5 ⊢ ((Fun Clsd ∧ (TopOn‘𝐵) ⊆ dom Clsd) → ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) → (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾}) ∈ (Clsd “ (TopOn‘𝐵))))
41	3, 39, 40	mp2an 705	. . . 4 ⊢ ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) → (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾}) ∈ (Clsd “ (TopOn‘𝐵)))
42	36, 41	syl 18	. . 3 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾}) ∈ (Clsd “ (TopOn‘𝐵)))
43	35, 42	eqeltrd 2865	. 2 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → 𝐾 ∈ (Clsd “ (TopOn‘𝐵)))
44	22, 43	impbii 212	1 ⊢ (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) ↔ (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  {crab 3418   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  dom cdm 5663   “ cima 5666  Fun wfun 6534   Fn wfn 6535  ‘cfv 6540  Moorecmre 17658  Topctop 23102  TopOnctopon 23119  Clsdccld 23225

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-mre 17662  df-top 23103  df-topon 23120  df-cld 23228

This theorem is used by: (None)