Theorem iscldtop 23118

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 23045	. . . . 5 ⊢ Clsd Fn Top
2		fnfun 6668	. . . . 5 ⊢ (Clsd Fn Top → Fun Clsd)
3	1, 2	ax-mp 5	. . . 4 ⊢ Fun Clsd
4		fvelima 6973	. . . 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 23117	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → (Clsd‘𝑎) ∈ (Moore‘𝐵))
7		topontop 22934	. . . . . . 7 ⊢ (𝑎 ∈ (TopOn‘𝐵) → 𝑎 ∈ Top)
8		0cld 23061	. . . . . . 7 ⊢ (𝑎 ∈ Top → ∅ ∈ (Clsd‘𝑎))
9	7, 8	syl 17	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ∅ ∈ (Clsd‘𝑎))
10		uncld 23064	. . . . . . . 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 3199	. . . . . 6 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎))
13	6, 9, 12	3jca 1127	. . . . 5 ⊢ (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ∧ ∅ ∈ (Clsd‘𝑎) ∧ ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥 ∪ 𝑦) ∈ (Clsd‘𝑎)))
14		eleq1 2826	. . . . . 6 ⊢ ((Clsd‘𝑎) = 𝐾 → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ↔ 𝐾 ∈ (Moore‘𝐵)))
15		eleq2 2827	. . . . . 6 ⊢ ((Clsd‘𝑎) = 𝐾 → (∅ ∈ (Clsd‘𝑎) ↔ ∅ ∈ 𝐾))
16		eleq2 2827	. . . . . . . 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 1435	. . . . 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 3145	. . 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 1135	. . . . 5 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → 𝐾 ∈ (Moore‘𝐵))
24		simp2 1136	. . . . 5 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → ∅ ∈ 𝐾)
25		uneq1 4170	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → (𝑥 ∪ 𝑦) = (𝑏 ∪ 𝑦))
26	25	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → ((𝑥 ∪ 𝑦) ∈ 𝐾 ↔ (𝑏 ∪ 𝑦) ∈ 𝐾))
27		uneq2 4171	. . . . . . . . . 10 ⊢ (𝑦 = 𝑐 → (𝑏 ∪ 𝑦) = (𝑏 ∪ 𝑐))
28	27	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑦 = 𝑐 → ((𝑏 ∪ 𝑦) ∈ 𝐾 ↔ (𝑏 ∪ 𝑐) ∈ 𝐾))
29	26, 28	rspc2v 3632	. . . . . . . 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 1134	. . . . . 6 ⊢ ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) → ((𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (𝑏 ∪ 𝑐) ∈ 𝐾))
32	31	3impib 1115	. . . . 5 ⊢ (((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝐾 (𝑥 ∪ 𝑦) ∈ 𝐾) ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) → (𝑏 ∪ 𝑐) ∈ 𝐾)
33		eqid 2734	. . . . 5 ⊢ {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾} = {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵 ∖ 𝑎) ∈ 𝐾}
34	23, 24, 32, 33	mretopd 23115	. . . 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 3998	. . . . . 6 ⊢ (TopOn‘𝐵) ⊆ Top
38	1	fndmi 6672	. . . . . 6 ⊢ dom Clsd = Top
39	37, 38	sseqtrri 4032	. . . . 5 ⊢ (TopOn‘𝐵) ⊆ dom Clsd
40		funfvima2 7250	. . . . 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 2838	. 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 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067  {crab 3432   ∖ cdif 3959   ∪ cun 3960   ⊆ wss 3962  ∅c0 4338  𝒫 cpw 4604  dom cdm 5688   “ cima 5691  Fun wfun 6556   Fn wfn 6557  ‘cfv 6562  Moorecmre 17626  Topctop 22914  TopOnctopon 22931  Clsdccld 23039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570  df-mre 17630  df-top 22915  df-topon 22932  df-cld 23042

This theorem is referenced by: (None)