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Theorem iscldtop 22449
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 β€œ (TopOnβ€˜π΅)) ↔ (𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾))
Distinct variable groups:   π‘₯,𝐡,𝑦   π‘₯,𝐾,𝑦

Proof of Theorem iscldtop
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fncld 22376 . . . . 5 Clsd Fn Top
2 fnfun 6603 . . . . 5 (Clsd Fn Top β†’ Fun Clsd)
31, 2ax-mp 5 . . . 4 Fun Clsd
4 fvelima 6909 . . . 4 ((Fun Clsd ∧ 𝐾 ∈ (Clsd β€œ (TopOnβ€˜π΅))) β†’ βˆƒπ‘Ž ∈ (TopOnβ€˜π΅)(Clsdβ€˜π‘Ž) = 𝐾)
53, 4mpan 689 . . 3 (𝐾 ∈ (Clsd β€œ (TopOnβ€˜π΅)) β†’ βˆƒπ‘Ž ∈ (TopOnβ€˜π΅)(Clsdβ€˜π‘Ž) = 𝐾)
6 cldmreon 22448 . . . . . 6 (π‘Ž ∈ (TopOnβ€˜π΅) β†’ (Clsdβ€˜π‘Ž) ∈ (Mooreβ€˜π΅))
7 topontop 22265 . . . . . . 7 (π‘Ž ∈ (TopOnβ€˜π΅) β†’ π‘Ž ∈ Top)
8 0cld 22392 . . . . . . 7 (π‘Ž ∈ Top β†’ βˆ… ∈ (Clsdβ€˜π‘Ž))
97, 8syl 17 . . . . . 6 (π‘Ž ∈ (TopOnβ€˜π΅) β†’ βˆ… ∈ (Clsdβ€˜π‘Ž))
10 uncld 22395 . . . . . . . 8 ((π‘₯ ∈ (Clsdβ€˜π‘Ž) ∧ 𝑦 ∈ (Clsdβ€˜π‘Ž)) β†’ (π‘₯ βˆͺ 𝑦) ∈ (Clsdβ€˜π‘Ž))
1110adantl 483 . . . . . . 7 ((π‘Ž ∈ (TopOnβ€˜π΅) ∧ (π‘₯ ∈ (Clsdβ€˜π‘Ž) ∧ 𝑦 ∈ (Clsdβ€˜π‘Ž))) β†’ (π‘₯ βˆͺ 𝑦) ∈ (Clsdβ€˜π‘Ž))
1211ralrimivva 3198 . . . . . 6 (π‘Ž ∈ (TopOnβ€˜π΅) β†’ βˆ€π‘₯ ∈ (Clsdβ€˜π‘Ž)βˆ€π‘¦ ∈ (Clsdβ€˜π‘Ž)(π‘₯ βˆͺ 𝑦) ∈ (Clsdβ€˜π‘Ž))
136, 9, 123jca 1129 . . . . 5 (π‘Ž ∈ (TopOnβ€˜π΅) β†’ ((Clsdβ€˜π‘Ž) ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ (Clsdβ€˜π‘Ž) ∧ βˆ€π‘₯ ∈ (Clsdβ€˜π‘Ž)βˆ€π‘¦ ∈ (Clsdβ€˜π‘Ž)(π‘₯ βˆͺ 𝑦) ∈ (Clsdβ€˜π‘Ž)))
14 eleq1 2826 . . . . . 6 ((Clsdβ€˜π‘Ž) = 𝐾 β†’ ((Clsdβ€˜π‘Ž) ∈ (Mooreβ€˜π΅) ↔ 𝐾 ∈ (Mooreβ€˜π΅)))
15 eleq2 2827 . . . . . 6 ((Clsdβ€˜π‘Ž) = 𝐾 β†’ (βˆ… ∈ (Clsdβ€˜π‘Ž) ↔ βˆ… ∈ 𝐾))
16 eleq2 2827 . . . . . . . 8 ((Clsdβ€˜π‘Ž) = 𝐾 β†’ ((π‘₯ βˆͺ 𝑦) ∈ (Clsdβ€˜π‘Ž) ↔ (π‘₯ βˆͺ 𝑦) ∈ 𝐾))
1716raleqbi1dv 3308 . . . . . . 7 ((Clsdβ€˜π‘Ž) = 𝐾 β†’ (βˆ€π‘¦ ∈ (Clsdβ€˜π‘Ž)(π‘₯ βˆͺ 𝑦) ∈ (Clsdβ€˜π‘Ž) ↔ βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾))
1817raleqbi1dv 3308 . . . . . 6 ((Clsdβ€˜π‘Ž) = 𝐾 β†’ (βˆ€π‘₯ ∈ (Clsdβ€˜π‘Ž)βˆ€π‘¦ ∈ (Clsdβ€˜π‘Ž)(π‘₯ βˆͺ 𝑦) ∈ (Clsdβ€˜π‘Ž) ↔ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾))
1914, 15, 183anbi123d 1437 . . . . 5 ((Clsdβ€˜π‘Ž) = 𝐾 β†’ (((Clsdβ€˜π‘Ž) ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ (Clsdβ€˜π‘Ž) ∧ βˆ€π‘₯ ∈ (Clsdβ€˜π‘Ž)βˆ€π‘¦ ∈ (Clsdβ€˜π‘Ž)(π‘₯ βˆͺ 𝑦) ∈ (Clsdβ€˜π‘Ž)) ↔ (𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾)))
2013, 19syl5ibcom 244 . . . 4 (π‘Ž ∈ (TopOnβ€˜π΅) β†’ ((Clsdβ€˜π‘Ž) = 𝐾 β†’ (𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾)))
2120rexlimiv 3146 . . 3 (βˆƒπ‘Ž ∈ (TopOnβ€˜π΅)(Clsdβ€˜π‘Ž) = 𝐾 β†’ (𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾))
225, 21syl 17 . 2 (𝐾 ∈ (Clsd β€œ (TopOnβ€˜π΅)) β†’ (𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾))
23 simp1 1137 . . . . 5 ((𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾) β†’ 𝐾 ∈ (Mooreβ€˜π΅))
24 simp2 1138 . . . . 5 ((𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾) β†’ βˆ… ∈ 𝐾)
25 uneq1 4117 . . . . . . . . . 10 (π‘₯ = 𝑏 β†’ (π‘₯ βˆͺ 𝑦) = (𝑏 βˆͺ 𝑦))
2625eleq1d 2823 . . . . . . . . 9 (π‘₯ = 𝑏 β†’ ((π‘₯ βˆͺ 𝑦) ∈ 𝐾 ↔ (𝑏 βˆͺ 𝑦) ∈ 𝐾))
27 uneq2 4118 . . . . . . . . . 10 (𝑦 = 𝑐 β†’ (𝑏 βˆͺ 𝑦) = (𝑏 βˆͺ 𝑐))
2827eleq1d 2823 . . . . . . . . 9 (𝑦 = 𝑐 β†’ ((𝑏 βˆͺ 𝑦) ∈ 𝐾 ↔ (𝑏 βˆͺ 𝑐) ∈ 𝐾))
2926, 28rspc2v 3591 . . . . . . . 8 ((𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) β†’ (βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾 β†’ (𝑏 βˆͺ 𝑐) ∈ 𝐾))
3029com12 32 . . . . . . 7 (βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾 β†’ ((𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) β†’ (𝑏 βˆͺ 𝑐) ∈ 𝐾))
31303ad2ant3 1136 . . . . . 6 ((𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾) β†’ ((𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) β†’ (𝑏 βˆͺ 𝑐) ∈ 𝐾))
32313impib 1117 . . . . 5 (((𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾) ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝐾) β†’ (𝑏 βˆͺ 𝑐) ∈ 𝐾)
33 eqid 2737 . . . . 5 {π‘Ž ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– π‘Ž) ∈ 𝐾} = {π‘Ž ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– π‘Ž) ∈ 𝐾}
3423, 24, 32, 33mretopd 22446 . . . 4 ((𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾) β†’ ({π‘Ž ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– π‘Ž) ∈ 𝐾} ∈ (TopOnβ€˜π΅) ∧ 𝐾 = (Clsdβ€˜{π‘Ž ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– π‘Ž) ∈ 𝐾})))
3534simprd 497 . . 3 ((𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾) β†’ 𝐾 = (Clsdβ€˜{π‘Ž ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– π‘Ž) ∈ 𝐾}))
3634simpld 496 . . . 4 ((𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾) β†’ {π‘Ž ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– π‘Ž) ∈ 𝐾} ∈ (TopOnβ€˜π΅))
377ssriv 3949 . . . . . 6 (TopOnβ€˜π΅) βŠ† Top
381fndmi 6607 . . . . . 6 dom Clsd = Top
3937, 38sseqtrri 3982 . . . . 5 (TopOnβ€˜π΅) βŠ† dom Clsd
40 funfvima2 7182 . . . . 5 ((Fun Clsd ∧ (TopOnβ€˜π΅) βŠ† dom Clsd) β†’ ({π‘Ž ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– π‘Ž) ∈ 𝐾} ∈ (TopOnβ€˜π΅) β†’ (Clsdβ€˜{π‘Ž ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– π‘Ž) ∈ 𝐾}) ∈ (Clsd β€œ (TopOnβ€˜π΅))))
413, 39, 40mp2an 691 . . . 4 ({π‘Ž ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– π‘Ž) ∈ 𝐾} ∈ (TopOnβ€˜π΅) β†’ (Clsdβ€˜{π‘Ž ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– π‘Ž) ∈ 𝐾}) ∈ (Clsd β€œ (TopOnβ€˜π΅)))
4236, 41syl 17 . . 3 ((𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾) β†’ (Clsdβ€˜{π‘Ž ∈ 𝒫 𝐡 ∣ (𝐡 βˆ– π‘Ž) ∈ 𝐾}) ∈ (Clsd β€œ (TopOnβ€˜π΅)))
4335, 42eqeltrd 2838 . 2 ((𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾) β†’ 𝐾 ∈ (Clsd β€œ (TopOnβ€˜π΅)))
4422, 43impbii 208 1 (𝐾 ∈ (Clsd β€œ (TopOnβ€˜π΅)) ↔ (𝐾 ∈ (Mooreβ€˜π΅) ∧ βˆ… ∈ 𝐾 ∧ βˆ€π‘₯ ∈ 𝐾 βˆ€π‘¦ ∈ 𝐾 (π‘₯ βˆͺ 𝑦) ∈ 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  βˆƒwrex 3074  {crab 3408   βˆ– cdif 3908   βˆͺ cun 3909   βŠ† wss 3911  βˆ…c0 4283  π’« cpw 4561  dom cdm 5634   β€œ cima 5637  Fun wfun 6491   Fn wfn 6492  β€˜cfv 6497  Moorecmre 17463  Topctop 22245  TopOnctopon 22262  Clsdccld 22370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505  df-mre 17467  df-top 22246  df-topon 22263  df-cld 22373
This theorem is referenced by: (None)
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