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Theorem iscldtop 20947
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 20874 . . . . 5 Clsd Fn Top
2 fnfun 6026 . . . . 5 (Clsd Fn Top → Fun Clsd)
31, 2ax-mp 5 . . . 4 Fun Clsd
4 fvelima 6287 . . . 4 ((Fun Clsd ∧ 𝐾 ∈ (Clsd “ (TopOn‘𝐵))) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
53, 4mpan 706 . . 3 (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
6 cldmreon 20946 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → (Clsd‘𝑎) ∈ (Moore‘𝐵))
7 topontop 20766 . . . . . . 7 (𝑎 ∈ (TopOn‘𝐵) → 𝑎 ∈ Top)
8 0cld 20890 . . . . . . 7 (𝑎 ∈ Top → ∅ ∈ (Clsd‘𝑎))
97, 8syl 17 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → ∅ ∈ (Clsd‘𝑎))
10 uncld 20893 . . . . . . . 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 3000 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎))
136, 9, 123jca 1261 . . . . 5 (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ∧ ∅ ∈ (Clsd‘𝑎) ∧ ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎)))
14 eleq1 2718 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ↔ 𝐾 ∈ (Moore‘𝐵)))
15 eleq2 2719 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → (∅ ∈ (Clsd‘𝑎) ↔ ∅ ∈ 𝐾))
16 eleq2 2719 . . . . . . . 8 ((Clsd‘𝑎) = 𝐾 → ((𝑥𝑦) ∈ (Clsd‘𝑎) ↔ (𝑥𝑦) ∈ 𝐾))
1716raleqbi1dv 3176 . . . . . . 7 ((Clsd‘𝑎) = 𝐾 → (∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
1817raleqbi1dv 3176 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → (∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
1914, 15, 183anbi123d 1439 . . . . 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 235 . . . 4 (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) = 𝐾 → (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾)))
2120rexlimiv 3056 . . 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 1081 . . . . 5 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 ∈ (Moore‘𝐵))
24 simp2 1082 . . . . 5 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ∅ ∈ 𝐾)
25 uneq1 3793 . . . . . . . . . 10 (𝑥 = 𝑏 → (𝑥𝑦) = (𝑏𝑦))
2625eleq1d 2715 . . . . . . . . 9 (𝑥 = 𝑏 → ((𝑥𝑦) ∈ 𝐾 ↔ (𝑏𝑦) ∈ 𝐾))
27 uneq2 3794 . . . . . . . . . 10 (𝑦 = 𝑐 → (𝑏𝑦) = (𝑏𝑐))
2827eleq1d 2715 . . . . . . . . 9 (𝑦 = 𝑐 → ((𝑏𝑦) ∈ 𝐾 ↔ (𝑏𝑐) ∈ 𝐾))
2926, 28rspc2v 3353 . . . . . . . 8 ((𝑏𝐾𝑐𝐾) → (∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾 → (𝑏𝑐) ∈ 𝐾))
3029com12 32 . . . . . . 7 (∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾 → ((𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾))
31303ad2ant3 1104 . . . . . 6 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ((𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾))
32313impib 1281 . . . . 5 (((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) ∧ 𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾)
33 eqid 2651 . . . . 5 {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} = {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}
3423, 24, 32, 33mretopd 20944 . . . 4 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) ∧ 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾})))
3534simprd 478 . . 3 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}))
3634simpld 474 . . . 4 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵))
377ssriv 3640 . . . . . 6 (TopOn‘𝐵) ⊆ Top
38 fndm 6028 . . . . . . 7 (Clsd Fn Top → dom Clsd = Top)
391, 38ax-mp 5 . . . . . 6 dom Clsd = Top
4037, 39sseqtr4i 3671 . . . . 5 (TopOn‘𝐵) ⊆ dom Clsd
41 funfvima2 6533 . . . . 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 708 . . . 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 2730 . 2 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 ∈ (Clsd “ (TopOn‘𝐵)))
4522, 44impbii 199 1 (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) ↔ (𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  cdif 3604  cun 3605  wss 3607  c0 3948  𝒫 cpw 4191  dom cdm 5143  cima 5146  Fun wfun 5920   Fn wfn 5921  cfv 5926  Moorecmre 16289  Topctop 20746  TopOnctopon 20763  Clsdccld 20868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-mre 16293  df-top 20747  df-topon 20764  df-cld 20871
This theorem is referenced by: (None)
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