Theorem iscldtop 22989

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 22916	. . . . 5 ⊢ Clsd Fn Top
2		fnfun 6621	. . . . 5 ⊢ (Clsd Fn Top → Fun Clsd)
3	1, 2	ax-mp 5	. . . 4 ⊢ Fun Clsd
4		fvelima 6929	. . . 4 ⊢ ((Fun Clsd ∧ 𝐾 ∈ (Clsd “ (TopOn‘𝐵))) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
5	3, 4	mpan 690	. . 3 ⊢ (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
6		cldmreon 22988	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → (Clsd‘𝑎) ∈ (Moore‘𝐵))
7		topontop 22807	. . . . . . 7 ⊢ (𝑎 ∈ (TopOn‘𝐵) → 𝑎 ∈ Top)
8		0cld 22932	. . . . . . 7 ⊢ (𝑎 ∈ Top → ∅ ∈ (Clsd‘𝑎))
9	7, 8	syl 17	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ∅ ∈ (Clsd‘𝑎))
10		uncld 22935	. . . . . . . 8 ⊢ ((𝑥 ∈ (Clsd‘𝑎) ∧ 𝑦 ∈ (Clsd‘𝑎)) → (𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎))
11	10	adantl 481	. . . . . . 7 ⊢ ((𝑎 ∈ (TopOn‘𝐵) ∧ (𝑥 ∈ (Clsd‘𝑎) ∧ 𝑦 ∈ (Clsd‘𝑎))) → (𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎))
12	11	ralrimivva 3181	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎))
13	6, 9, 12	3jca 1128	. . . . 5 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ∧ ∅ ∈ (Clsd‘𝑎) ∧ ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎)))
14		eleq1 2817	. . . . . 6 ⊢ ((Clsd‘𝑎) = 𝐾 → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ↔ 𝐾 ∈ (Moore‘𝐵)))
15		eleq2 2818	. . . . . 6 ⊢ ((Clsd‘𝑎) = 𝐾 → (∅ ∈ (Clsd‘𝑎) ↔ ∅ ∈ 𝐾))
16		eleq2 2818	. . . . . . . 8 ⊢ ((Clsd‘𝑎) = 𝐾 → ((𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎) ↔ (𝑥 ∪ 𝑦) ∈ 𝐾))
17	16	raleqbi1dv 3313	. . . . . . 7 ⊢ ((Clsd‘𝑎) = 𝐾 → (∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))
18	17	raleqbi1dv 3313	. . . . . 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 1438	. . . . 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 245	. . . 4 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) = 𝐾 → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾)))
21	20	rexlimiv 3128	. . 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 1136	. . . . 5 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → 𝐾 ∈ (Moore‘𝐵))
24		simp2 1137	. . . . 5 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → ∅ ∈ 𝐾)
25		uneq1 4127	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (𝑥 ∪ 𝑦) = (𝑏 ∪ 𝑦))
26	25	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → ((𝑥 ∪ 𝑦) ∈ 𝐾 ↔ (𝑏 ∪ 𝑦) ∈ 𝐾))
27		uneq2 4128	. . . . . . . . . 10 ⊢ (𝑦 = 𝑐 → (𝑏 ∪ 𝑦) = (𝑏 ∪ 𝑐))
28	27	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑦 = 𝑐 → ((𝑏 ∪ 𝑦) ∈ 𝐾 ↔ (𝑏 ∪ 𝑐) ∈ 𝐾))
29	26, 28	rspc2v 3602	. . . . . . . 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 1135	. . . . . 6 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → ((𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (𝑏 ∪ 𝑐) ∈ 𝐾))
32	31	3impib 1116	. . . . 5 ⊢ (((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (𝑏 ∪ 𝑐) ∈ 𝐾)
33		eqid 2730	. . . . 5 ⊢ {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾} = {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾}
34	23, 24, 32, 33	mretopd 22986	. . . 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 495	. . 3 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾}))
36	34	simpld 494	. . . 4 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵))
37	7	ssriv 3953	. . . . . 6 ⊢ (TopOn‘𝐵) ⊆ Top
38	1	fndmi 6625	. . . . . 6 ⊢ dom Clsd = Top
39	37, 38	sseqtrri 3999	. . . . 5 ⊢ (TopOn‘𝐵) ⊆ dom Clsd
40		funfvima2 7208	. . . . 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 692	. . . 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 2829	. 2 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → 𝐾 ∈ (Clsd “ (TopOn‘𝐵)))
44	22, 43	impbii 209	1 ⊢ (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) ↔ (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  {crab 3408   ∖ cdif 3914   ∪ cun 3915   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  dom cdm 5641   “ cima 5644  Fun wfun 6508   Fn wfn 6509  ‘cfv 6514  Moorecmre 17550  Topctop 22787  TopOnctopon 22804  Clsdccld 22910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522  df-mre 17554  df-top 22788  df-topon 22805  df-cld 22913

This theorem is referenced by: (None)