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Theorem iscldtop 20809
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 20736 . . . . 5 Clsd Fn Top
2 fnfun 5946 . . . . 5 (Clsd Fn Top → Fun Clsd)
31, 2ax-mp 5 . . . 4 Fun Clsd
4 fvelima 6205 . . . 4 ((Fun Clsd ∧ 𝐾 ∈ (Clsd “ (TopOn‘𝐵))) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
53, 4mpan 705 . . 3 (𝐾 ∈ (Clsd “ (TopOn‘𝐵)) → ∃𝑎 ∈ (TopOn‘𝐵)(Clsd‘𝑎) = 𝐾)
6 cldmreon 20808 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → (Clsd‘𝑎) ∈ (Moore‘𝐵))
7 topontop 20641 . . . . . . 7 (𝑎 ∈ (TopOn‘𝐵) → 𝑎 ∈ Top)
8 0cld 20752 . . . . . . 7 (𝑎 ∈ Top → ∅ ∈ (Clsd‘𝑎))
97, 8syl 17 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → ∅ ∈ (Clsd‘𝑎))
10 uncld 20755 . . . . . . . 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 2965 . . . . . 6 (𝑎 ∈ (TopOn‘𝐵) → ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎))
136, 9, 123jca 1240 . . . . 5 (𝑎 ∈ (TopOn‘𝐵) → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ∧ ∅ ∈ (Clsd‘𝑎) ∧ ∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎)))
14 eleq1 2686 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → ((Clsd‘𝑎) ∈ (Moore‘𝐵) ↔ 𝐾 ∈ (Moore‘𝐵)))
15 eleq2 2687 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → (∅ ∈ (Clsd‘𝑎) ↔ ∅ ∈ 𝐾))
16 eleq2 2687 . . . . . . . 8 ((Clsd‘𝑎) = 𝐾 → ((𝑥𝑦) ∈ (Clsd‘𝑎) ↔ (𝑥𝑦) ∈ 𝐾))
1716raleqbi1dv 3135 . . . . . . 7 ((Clsd‘𝑎) = 𝐾 → (∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
1817raleqbi1dv 3135 . . . . . 6 ((Clsd‘𝑎) = 𝐾 → (∀𝑥 ∈ (Clsd‘𝑎)∀𝑦 ∈ (Clsd‘𝑎)(𝑥𝑦) ∈ (Clsd‘𝑎) ↔ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾))
1914, 15, 183anbi123d 1396 . . . . 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 3020 . . 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 1059 . . . . 5 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 ∈ (Moore‘𝐵))
24 simp2 1060 . . . . 5 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ∅ ∈ 𝐾)
25 uneq1 3738 . . . . . . . . . 10 (𝑥 = 𝑏 → (𝑥𝑦) = (𝑏𝑦))
2625eleq1d 2683 . . . . . . . . 9 (𝑥 = 𝑏 → ((𝑥𝑦) ∈ 𝐾 ↔ (𝑏𝑦) ∈ 𝐾))
27 uneq2 3739 . . . . . . . . . 10 (𝑦 = 𝑐 → (𝑏𝑦) = (𝑏𝑐))
2827eleq1d 2683 . . . . . . . . 9 (𝑦 = 𝑐 → ((𝑏𝑦) ∈ 𝐾 ↔ (𝑏𝑐) ∈ 𝐾))
2926, 28rspc2v 3306 . . . . . . . 8 ((𝑏𝐾𝑐𝐾) → (∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾 → (𝑏𝑐) ∈ 𝐾))
3029com12 32 . . . . . . 7 (∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾 → ((𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾))
31303ad2ant3 1082 . . . . . 6 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ((𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾))
32313impib 1259 . . . . 5 (((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) ∧ 𝑏𝐾𝑐𝐾) → (𝑏𝑐) ∈ 𝐾)
33 eqid 2621 . . . . 5 {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} = {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}
3423, 24, 32, 33mretopd 20806 . . . 4 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → ({𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵) ∧ 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾})))
3534simprd 479 . . 3 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → 𝐾 = (Clsd‘{𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾}))
3634simpld 475 . . . 4 ((𝐾 ∈ (Moore‘𝐵) ∧ ∅ ∈ 𝐾 ∧ ∀𝑥𝐾𝑦𝐾 (𝑥𝑦) ∈ 𝐾) → {𝑎 ∈ 𝒫 𝐵 ∣ (𝐵𝑎) ∈ 𝐾} ∈ (TopOn‘𝐵))
377ssriv 3587 . . . . . 6 (TopOn‘𝐵) ⊆ Top
38 fndm 5948 . . . . . . 7 (Clsd Fn Top → dom Clsd = Top)
391, 38ax-mp 5 . . . . . 6 dom Clsd = Top
4037, 39sseqtr4i 3617 . . . . 5 (TopOn‘𝐵) ⊆ dom Clsd
41 funfvima2 6447 . . . . 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 707 . . . 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 2698 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  cdif 3552  cun 3553  wss 3555  c0 3891  𝒫 cpw 4130  dom cdm 5074  cima 5077  Fun wfun 5841   Fn wfn 5842  cfv 5847  Moorecmre 16163  Topctop 20617  TopOnctopon 20618  Clsdccld 20730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-mre 16167  df-top 20621  df-topon 20623  df-cld 20733
This theorem is referenced by: (None)
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