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Theorem iscldtop 23103
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 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 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.
StepHypRef Expression
1 fncld 23030 . . . . 5 Clsd Fn Top
2 fnfun 6668 . . . . 5 (Clsd Fn Top → Fun Clsd)
31, 2ax-mp 5 . . . 4 Fun Clsd
4 fvelima 6974 . . . 4 ((Fun Clsd ∧ 𝐾 ∈ (Clsd “ (TopOn‘𝐵))) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
53, 4mpan 690 . . 3 (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
6 cldmreon 23102 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → (Clsd‘𝑎) ∈ (Moore‘𝐵))
7 topontop 22919 . . . . . . 7 (𝑎 ∈ (TopOn‘𝐵) → 𝑎 ∈ Top)
8 0cld 23046 . . . . . . 7 (𝑎 ∈ Top → ∅ ∈ (Clsd‘𝑎))
97, 8syl 17 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → ∅ ∈ (Clsd‘𝑎))
10 uncld 23049 . . . . . . . 8 ((𝑥 ∈ (Clsd‘𝑎) ∧ 𝑦 ∈ (Clsd‘𝑎)) → (𝑥𝑦) ∈ (Clsd‘𝑎))
1110adantl 481 . . . . . . 7 ((𝑎 ∈ (TopOn‘𝐵) ∧ (𝑥 ∈ (Clsd‘𝑎) ∧ 𝑦 ∈ (Clsd‘𝑎))) → (𝑥𝑦) ∈ (Clsd‘𝑎))
1211ralrimivva 3202 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎))
136, 9, 123jca 1129 . . . . 5 (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ∧ ∅ ∈ (Clsd‘𝑎) ∧ ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎)))
14 eleq1 2829 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ↔ 𝐾 ∈ (Moore‘𝐵)))
15 eleq2 2830 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → (∅ ∈ (Clsd‘𝑎) ↔ ∅ ∈ 𝐾))
16 eleq2 2830 . . . . . . . 8 ((Clsd‘𝑎) = 𝐾 → ((𝑥𝑦) ∈ (Clsd‘𝑎) ↔ (𝑥𝑦) ∈ 𝐾))
1716raleqbi1dv 3338 . . . . . . 7 ((Clsd‘𝑎) = 𝐾 → (∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
1817raleqbi1dv 3338 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → (∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
1914, 15, 183anbi123d 1438 . . . . 5 ((Clsd‘𝑎) = 𝐾 → (((Clsd‘𝑎) ∈ (Moore‘𝐵) ∧ ∅ ∈ (Clsd‘𝑎) ∧ ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎)) ↔ (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾)))
2013, 19syl5ibcom 245 . . . 4 (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) = 𝐾 → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾)))
2120rexlimiv 3148 . . 3 (∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾 → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
225, 21syl 17 . 2 (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
23 simp1 1137 . . . . 5 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 ∈ (Moore‘𝐵))
24 simp2 1138 . . . . 5 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ∅ ∈ 𝐾)
25 uneq1 4161 . . . . . . . . . 10 (𝑥 = 𝑏 → (𝑥𝑦) = (𝑏𝑦))
2625eleq1d 2826 . . . . . . . . 9 (𝑥 = 𝑏 → ((𝑥𝑦) ∈ 𝐾 ↔ (𝑏𝑦) ∈ 𝐾))
27 uneq2 4162 . . . . . . . . . 10 (𝑦 = 𝑐 → (𝑏𝑦) = (𝑏𝑐))
2827eleq1d 2826 . . . . . . . . 9 (𝑦 = 𝑐 → ((𝑏𝑦) ∈ 𝐾 ↔ (𝑏𝑐) ∈ 𝐾))
2926, 28rspc2v 3633 . . . . . . . 8 ((𝑏𝐾𝑐𝐾) → (∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾 → (𝑏𝑐) ∈ 𝐾))
3029com12 32 . . . . . . 7 (∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾 → ((𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾))
31303ad2ant3 1136 . . . . . 6 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ((𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾))
32313impib 1117 . . . . 5 (((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) ∧ 𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾)
33 eqid 2737 . . . . 5 {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} = {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}
3423, 24, 32, 33mretopd 23100 . . . 4 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) ∧ 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾})))
3534simprd 495 . . 3 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}))
3634simpld 494 . . . 4 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵))
377ssriv 3987 . . . . . 6 (TopOn‘𝐵) ⊆ Top
381fndmi 6672 . . . . . 6 dom Clsd = Top
3937, 38sseqtrri 4033 . . . . 5 (TopOn‘𝐵) ⊆ dom Clsd
40 funfvima2 7251 . . . . 5 ((Fun Clsd ∧ (TopOn‘𝐵) ⊆ dom Clsd) → ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) → (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}) ∈ (Clsd “ (TopOn‘𝐵))))
413, 39, 40mp2an 692 . . . 4 ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) → (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}) ∈ (Clsd “ (TopOn‘𝐵)))
4236, 41syl 17 . . 3 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}) ∈ (Clsd “ (TopOn‘𝐵)))
4335, 42eqeltrd 2841 . 2 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 ∈ (Clsd “ (TopOn‘𝐵)))
4422, 43impbii 209 1 (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) ↔ (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  cdif 3948  cun 3949  wss 3951  c0 4333  𝒫 cpw 4600  dom cdm 5685  cima 5688  Fun wfun 6555   Fn wfn 6556  cfv 6561  Moorecmre 17625  Topctop 22899  TopOnctopon 22916  Clsdccld 23024
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-mre 17629  df-top 22900  df-topon 22917  df-cld 23027
This theorem is referenced by: (None)
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