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Theorem iscldtop 21631
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 21558 . . . . 5 Clsd Fn Top
2 fnfun 6446 . . . . 5 (Clsd Fn Top → Fun Clsd)
31, 2ax-mp 5 . . . 4 Fun Clsd
4 fvelima 6724 . . . 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 21630 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → (Clsd‘𝑎) ∈ (Moore‘𝐵))
7 topontop 21449 . . . . . . 7 (𝑎 ∈ (TopOn‘𝐵) → 𝑎 ∈ Top)
8 0cld 21574 . . . . . . 7 (𝑎 ∈ Top → ∅ ∈ (Clsd‘𝑎))
97, 8syl 17 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → ∅ ∈ (Clsd‘𝑎))
10 uncld 21577 . . . . . . . 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 3188 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎))
136, 9, 123jca 1120 . . . . 5 (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ∧ ∅ ∈ (Clsd‘𝑎) ∧ ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎)))
14 eleq1 2897 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ↔ 𝐾 ∈ (Moore‘𝐵)))
15 eleq2 2898 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → (∅ ∈ (Clsd‘𝑎) ↔ ∅ ∈ 𝐾))
16 eleq2 2898 . . . . . . . 8 ((Clsd‘𝑎) = 𝐾 → ((𝑥𝑦) ∈ (Clsd‘𝑎) ↔ (𝑥𝑦) ∈ 𝐾))
1716raleqbi1dv 3401 . . . . . . 7 ((Clsd‘𝑎) = 𝐾 → (∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
1817raleqbi1dv 3401 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → (∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
1914, 15, 183anbi123d 1427 . . . . 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 3277 . . 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 1128 . . . . 5 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 ∈ (Moore‘𝐵))
24 simp2 1129 . . . . 5 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ∅ ∈ 𝐾)
25 uneq1 4129 . . . . . . . . . 10 (𝑥 = 𝑏 → (𝑥𝑦) = (𝑏𝑦))
2625eleq1d 2894 . . . . . . . . 9 (𝑥 = 𝑏 → ((𝑥𝑦) ∈ 𝐾 ↔ (𝑏𝑦) ∈ 𝐾))
27 uneq2 4130 . . . . . . . . . 10 (𝑦 = 𝑐 → (𝑏𝑦) = (𝑏𝑐))
2827eleq1d 2894 . . . . . . . . 9 (𝑦 = 𝑐 → ((𝑏𝑦) ∈ 𝐾 ↔ (𝑏𝑐) ∈ 𝐾))
2926, 28rspc2v 3630 . . . . . . . 8 ((𝑏𝐾𝑐𝐾) → (∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾 → (𝑏𝑐) ∈ 𝐾))
3029com12 32 . . . . . . 7 (∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾 → ((𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾))
31303ad2ant3 1127 . . . . . 6 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ((𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾))
32313impib 1108 . . . . 5 (((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) ∧ 𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾)
33 eqid 2818 . . . . 5 {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} = {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}
3423, 24, 32, 33mretopd 21628 . . . 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 3968 . . . . . 6 (TopOn‘𝐵) ⊆ Top
38 fndm 6448 . . . . . . 7 (Clsd Fn Top → dom Clsd = Top)
391, 38ax-mp 5 . . . . . 6 dom Clsd = Top
4037, 39sseqtrri 4001 . . . . 5 (TopOn‘𝐵) ⊆ dom Clsd
41 funfvima2 6984 . . . . 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 2910 . 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 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  cdif 3930  cun 3931  wss 3933  c0 4288  𝒫 cpw 4535  dom cdm 5548  cima 5551  Fun wfun 6342   Fn wfn 6343  cfv 6348  Moorecmre 16841  Topctop 21429  TopOnctopon 21446  Clsdccld 21552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-mre 16845  df-top 21430  df-topon 21447  df-cld 21555
This theorem is referenced by: (None)
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