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Theorem iscldtop 21387
 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 21314 . . . . 5 Clsd Fn Top
2 fnfun 6323 . . . . 5 (Clsd Fn Top → Fun Clsd)
31, 2ax-mp 5 . . . 4 Fun Clsd
4 fvelima 6599 . . . 4 ((Fun Clsd ∧ 𝐾 ∈ (Clsd “ (TopOn‘𝐵))) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
53, 4mpan 686 . . 3 (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
6 cldmreon 21386 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → (Clsd‘𝑎) ∈ (Moore‘𝐵))
7 topontop 21205 . . . . . . 7 (𝑎 ∈ (TopOn‘𝐵) → 𝑎 ∈ Top)
8 0cld 21330 . . . . . . 7 (𝑎 ∈ Top → ∅ ∈ (Clsd‘𝑎))
97, 8syl 17 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → ∅ ∈ (Clsd‘𝑎))
10 uncld 21333 . . . . . . . 8 ((𝑥 ∈ (Clsd‘𝑎) ∧ 𝑦 ∈ (Clsd‘𝑎)) → (𝑥𝑦) ∈ (Clsd‘𝑎))
1110adantl 482 . . . . . . 7 ((𝑎 ∈ (TopOn‘𝐵) ∧ (𝑥 ∈ (Clsd‘𝑎) ∧ 𝑦 ∈ (Clsd‘𝑎))) → (𝑥𝑦) ∈ (Clsd‘𝑎))
1211ralrimivva 3158 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎))
136, 9, 123jca 1121 . . . . 5 (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ∧ ∅ ∈ (Clsd‘𝑎) ∧ ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎)))
14 eleq1 2870 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ↔ 𝐾 ∈ (Moore‘𝐵)))
15 eleq2 2871 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → (∅ ∈ (Clsd‘𝑎) ↔ ∅ ∈ 𝐾))
16 eleq2 2871 . . . . . . . 8 ((Clsd‘𝑎) = 𝐾 → ((𝑥𝑦) ∈ (Clsd‘𝑎) ↔ (𝑥𝑦) ∈ 𝐾))
1716raleqbi1dv 3363 . . . . . . 7 ((Clsd‘𝑎) = 𝐾 → (∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
1817raleqbi1dv 3363 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → (∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
1914, 15, 183anbi123d 1428 . . . . 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 246 . . . 4 (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) = 𝐾 → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾)))
2120rexlimiv 3243 . . 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 1129 . . . . 5 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 ∈ (Moore‘𝐵))
24 simp2 1130 . . . . 5 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ∅ ∈ 𝐾)
25 uneq1 4053 . . . . . . . . . 10 (𝑥 = 𝑏 → (𝑥𝑦) = (𝑏𝑦))
2625eleq1d 2867 . . . . . . . . 9 (𝑥 = 𝑏 → ((𝑥𝑦) ∈ 𝐾 ↔ (𝑏𝑦) ∈ 𝐾))
27 uneq2 4054 . . . . . . . . . 10 (𝑦 = 𝑐 → (𝑏𝑦) = (𝑏𝑐))
2827eleq1d 2867 . . . . . . . . 9 (𝑦 = 𝑐 → ((𝑏𝑦) ∈ 𝐾 ↔ (𝑏𝑐) ∈ 𝐾))
2926, 28rspc2v 3572 . . . . . . . 8 ((𝑏𝐾𝑐𝐾) → (∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾 → (𝑏𝑐) ∈ 𝐾))
3029com12 32 . . . . . . 7 (∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾 → ((𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾))
31303ad2ant3 1128 . . . . . 6 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ((𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾))
32313impib 1109 . . . . 5 (((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) ∧ 𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾)
33 eqid 2795 . . . . 5 {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} = {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}
3423, 24, 32, 33mretopd 21384 . . . 4 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) ∧ 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾})))
3534simprd 496 . . 3 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}))
3634simpld 495 . . . 4 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵))
377ssriv 3893 . . . . . 6 (TopOn‘𝐵) ⊆ Top
38 fndm 6325 . . . . . . 7 (Clsd Fn Top → dom Clsd = Top)
391, 38ax-mp 5 . . . . . 6 dom Clsd = Top
4037, 39sseqtr4i 3925 . . . . 5 (TopOn‘𝐵) ⊆ dom Clsd
41 funfvima2 6859 . . . . 5 ((Fun Clsd ∧ (TopOn‘𝐵) ⊆ dom Clsd) → ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) → (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}) ∈ (Clsd “ (TopOn‘𝐵))))
423, 40, 41mp2an 688 . . . 4 ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) → (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}) ∈ (Clsd “ (TopOn‘𝐵)))
4336, 42syl 17 . . 3 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}) ∈ (Clsd “ (TopOn‘𝐵)))
4435, 43eqeltrd 2883 . 2 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 ∈ (Clsd “ (TopOn‘𝐵)))
4522, 44impbii 210 1 (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) ↔ (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  ∀wral 3105  ∃wrex 3106  {crab 3109   ∖ cdif 3856   ∪ cun 3857   ⊆ wss 3859  ∅c0 4211  𝒫 cpw 4453  dom cdm 5443   “ cima 5446  Fun wfun 6219   Fn wfn 6220  ‘cfv 6225  Moorecmre 16682  Topctop 21185  TopOnctopon 21202  Clsdccld 21308 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-fv 6233  df-mre 16686  df-top 21186  df-topon 21203  df-cld 21311 This theorem is referenced by: (None)
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