Theorem iscldtop 23078

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 23005	. . . . 5 ⊢ Clsd Fn Top
2		fnfun 6585	. . . . 5 ⊢ (Clsd Fn Top → Fun Clsd)
3	1, 2	ax-mp 5	. . . 4 ⊢ Fun Clsd
4		fvelima 6892	. . . 4 ⊢ ((Fun Clsd ∧ 𝐾 ∈ (Clsd “ (TopOn‘𝐵))) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
5	3, 4	mpan 696	. . 3 ⊢ (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
6		cldmreon 23077	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → (Clsd‘𝑎) ∈ (Moore‘𝐵))
7		topontop 22896	. . . . . . 7 ⊢ (𝑎 ∈ (TopOn‘𝐵) → 𝑎 ∈ Top)
8		0cld 23021	. . . . . . 7 ⊢ (𝑎 ∈ Top → ∅ ∈ (Clsd‘𝑎))
9	7, 8	syl 17	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ∅ ∈ (Clsd‘𝑎))
10		uncld 23024	. . . . . . . 8 ⊢ ((𝑥 ∈ (Clsd‘𝑎) ∧ 𝑦 ∈ (Clsd‘𝑎)) → (𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎))
11	10	adantl 482	. . . . . . 7 ⊢ ((𝑎 ∈ (TopOn‘𝐵) ∧ (𝑥 ∈ (Clsd‘𝑎) ∧ 𝑦 ∈ (Clsd‘𝑎))) → (𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎))
12	11	ralrimivva 3182	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎))
13	6, 9, 12	3jca 1134	. . . . 5 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ∧ ∅ ∈ (Clsd‘𝑎) ∧ ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎)))
14		eleq1 2827	. . . . . 6 ⊢ ((Clsd‘𝑎) = 𝐾 → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ↔ 𝐾 ∈ (Moore‘𝐵)))
15		eleq2 2828	. . . . . 6 ⊢ ((Clsd‘𝑎) = 𝐾 → (∅ ∈ (Clsd‘𝑎) ↔ ∅ ∈ 𝐾))
16		eleq2 2828	. . . . . . . 8 ⊢ ((Clsd‘𝑎) = 𝐾 → ((𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎) ↔ (𝑥 ∪ 𝑦) ∈ 𝐾))
17	16	raleqbi1dv 3307	. . . . . . 7 ⊢ ((Clsd‘𝑎) = 𝐾 → (∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))
18	17	raleqbi1dv 3307	. . . . . 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 1444	. . . . 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 246	. . . 4 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) = 𝐾 → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾)))
21	20	rexlimiv 3133	. . 3 ⊢ (∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾 → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))
22	5, 21	syl 17	. 2 ⊢ (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))
23		simp1 1142	. . . . 5 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → 𝐾 ∈ (Moore‘𝐵))
24		simp2 1143	. . . . 5 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → ∅ ∈ 𝐾)
25		uneq1 4091	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (𝑥 ∪ 𝑦) = (𝑏 ∪ 𝑦))
26	25	eleq1d 2824	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → ((𝑥 ∪ 𝑦) ∈ 𝐾 ↔ (𝑏 ∪ 𝑦) ∈ 𝐾))
27		uneq2 4092	. . . . . . . . . 10 ⊢ (𝑦 = 𝑐 → (𝑏 ∪ 𝑦) = (𝑏 ∪ 𝑐))
28	27	eleq1d 2824	. . . . . . . . 9 ⊢ (𝑦 = 𝑐 → ((𝑏 ∪ 𝑦) ∈ 𝐾 ↔ (𝑏 ∪ 𝑐) ∈ 𝐾))
29	26, 28	rspc2v 3571	. . . . . . . 8 ⊢ ((𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾 → (𝑏 ∪ 𝑐) ∈ 𝐾))
30	29	com12 32	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾 → ((𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (𝑏 ∪ 𝑐) ∈ 𝐾))
31	30	3ad2ant3 1141	. . . . . 6 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → ((𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (𝑏 ∪ 𝑐) ∈ 𝐾))
32	31	3impib 1122	. . . . 5 ⊢ (((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (𝑏 ∪ 𝑐) ∈ 𝐾)
33		eqid 2739	. . . . 5 ⊢ {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾} = {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾}
34	23, 24, 32, 33	mretopd 23075	. . . 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 496	. . 3 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾}))
36	34	simpld 495	. . . 4 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵))
37	7	ssriv 3919	. . . . . 6 ⊢ (TopOn‘𝐵) ⊆ Top
38	1	fndmi 6589	. . . . . 6 ⊢ dom Clsd = Top
39	37, 38	sseqtrri 3964	. . . . 5 ⊢ (TopOn‘𝐵) ⊆ dom Clsd
40		funfvima2 7175	. . . . 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 698	. . . 4 ⊢ ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) → (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾}) ∈ (Clsd “ (TopOn‘𝐵)))
42	36, 41	syl 17	. . 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 2839	. 2 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → 𝐾 ∈ (Clsd “ (TopOn‘𝐵)))
44	22, 43	impbii 210	1 ⊢ (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) ↔ (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  {crab 3391   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  dom cdm 5618   “ cima 5621  Fun wfun 6479   Fn wfn 6480  ‘cfv 6485  Moorecmre 17535  Topctop 22876  TopOnctopon 22893  Clsdccld 22999

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 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-fv 6493  df-mre 17539  df-top 22877  df-topon 22894  df-cld 23002

This theorem is referenced by: (None)