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Theorem istopclsd 37765
Description: A closure function which satisfies sscls 21062, clsidm 21073, cls0 21086, and clsun 32629 defines a (unique) topology which it is the closure function on. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
istopclsd.b (𝜑𝐵𝑉)
istopclsd.f (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)
istopclsd.e ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))
istopclsd.i ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
istopclsd.z (𝜑 → (𝐹‘∅) = ∅)
istopclsd.u ((𝜑𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))
istopclsd.j 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)}
Assertion
Ref Expression
istopclsd (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))
Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐽,𝑦   𝑥,𝑉,𝑦,𝑧
Allowed substitution hint:   𝐽(𝑧)

Proof of Theorem istopclsd
StepHypRef Expression
1 istopclsd.j . . . 4 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)}
2 istopclsd.f . . . . . . . . 9 (𝜑𝐹:𝒫 𝐵⟶𝒫 𝐵)
3 ffn 6206 . . . . . . . . 9 (𝐹:𝒫 𝐵⟶𝒫 𝐵𝐹 Fn 𝒫 𝐵)
42, 3syl 17 . . . . . . . 8 (𝜑𝐹 Fn 𝒫 𝐵)
54adantr 472 . . . . . . 7 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
6 difss 3880 . . . . . . . . 9 (𝐵𝑧) ⊆ 𝐵
7 istopclsd.b . . . . . . . . . 10 (𝜑𝐵𝑉)
8 elpw2g 4976 . . . . . . . . . 10 (𝐵𝑉 → ((𝐵𝑧) ∈ 𝒫 𝐵 ↔ (𝐵𝑧) ⊆ 𝐵))
97, 8syl 17 . . . . . . . . 9 (𝜑 → ((𝐵𝑧) ∈ 𝒫 𝐵 ↔ (𝐵𝑧) ⊆ 𝐵))
106, 9mpbiri 248 . . . . . . . 8 (𝜑 → (𝐵𝑧) ∈ 𝒫 𝐵)
1110adantr 472 . . . . . . 7 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐵𝑧) ∈ 𝒫 𝐵)
12 fnelfp 6605 . . . . . . 7 ((𝐹 Fn 𝒫 𝐵 ∧ (𝐵𝑧) ∈ 𝒫 𝐵) → ((𝐵𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)))
135, 11, 12syl2anc 696 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((𝐵𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)))
1413bicomd 213 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((𝐹‘(𝐵𝑧)) = (𝐵𝑧) ↔ (𝐵𝑧) ∈ dom (𝐹 ∩ I )))
1514rabbidva 3328 . . . 4 (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )})
161, 15syl5eq 2806 . . 3 (𝜑𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )})
17 istopclsd.e . . . . . 6 ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))
18 simp1 1131 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → 𝜑)
19 simp2 1132 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → 𝑥𝐵)
20 simp3 1133 . . . . . . . . . 10 ((𝜑𝑥𝐵𝑦𝑥) → 𝑦𝑥)
2120, 19sstrd 3754 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → 𝑦𝐵)
22 istopclsd.u . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))
2318, 19, 21, 22syl3anc 1477 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝑥) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))
24 ssequn2 3929 . . . . . . . . . . 11 (𝑦𝑥 ↔ (𝑥𝑦) = 𝑥)
2524biimpi 206 . . . . . . . . . 10 (𝑦𝑥 → (𝑥𝑦) = 𝑥)
26253ad2ant3 1130 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → (𝑥𝑦) = 𝑥)
2726fveq2d 6356 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝑥) → (𝐹‘(𝑥𝑦)) = (𝐹𝑥))
2823, 27eqtr3d 2796 . . . . . . 7 ((𝜑𝑥𝐵𝑦𝑥) → ((𝐹𝑥) ∪ (𝐹𝑦)) = (𝐹𝑥))
29 ssequn2 3929 . . . . . . 7 ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ ((𝐹𝑥) ∪ (𝐹𝑦)) = (𝐹𝑥))
3028, 29sylibr 224 . . . . . 6 ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))
31 istopclsd.i . . . . . 6 ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
327, 2, 17, 30, 31ismrcd1 37763 . . . . 5 (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
33 istopclsd.z . . . . . 6 (𝜑 → (𝐹‘∅) = ∅)
34 0elpw 4983 . . . . . . 7 ∅ ∈ 𝒫 𝐵
35 fnelfp 6605 . . . . . . 7 ((𝐹 Fn 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
364, 34, 35sylancl 697 . . . . . 6 (𝜑 → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
3733, 36mpbird 247 . . . . 5 (𝜑 → ∅ ∈ dom (𝐹 ∩ I ))
38 simp1 1131 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝜑)
39 inss1 3976 . . . . . . . . . . . . 13 (𝐹 ∩ I ) ⊆ 𝐹
40 dmss 5478 . . . . . . . . . . . . 13 ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
4139, 40ax-mp 5 . . . . . . . . . . . 12 dom (𝐹 ∩ I ) ⊆ dom 𝐹
42 fdm 6212 . . . . . . . . . . . . 13 (𝐹:𝒫 𝐵⟶𝒫 𝐵 → dom 𝐹 = 𝒫 𝐵)
432, 42syl 17 . . . . . . . . . . . 12 (𝜑 → dom 𝐹 = 𝒫 𝐵)
4441, 43syl5sseq 3794 . . . . . . . . . . 11 (𝜑 → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
45443ad2ant1 1128 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
46 simp2 1132 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ dom (𝐹 ∩ I ))
4745, 46sseldd 3745 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ 𝒫 𝐵)
4847elpwid 4314 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥𝐵)
49 simp3 1133 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ dom (𝐹 ∩ I ))
5045, 49sseldd 3745 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ 𝒫 𝐵)
5150elpwid 4314 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦𝐵)
5238, 48, 51, 22syl3anc 1477 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))
5343ad2ant1 1128 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝐹 Fn 𝒫 𝐵)
54 fnelfp 6605 . . . . . . . . . 10 ((𝐹 Fn 𝒫 𝐵𝑥 ∈ 𝒫 𝐵) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑥) = 𝑥))
5553, 47, 54syl2anc 696 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑥) = 𝑥))
5646, 55mpbid 222 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹𝑥) = 𝑥)
57 fnelfp 6605 . . . . . . . . . 10 ((𝐹 Fn 𝒫 𝐵𝑦 ∈ 𝒫 𝐵) → (𝑦 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑦) = 𝑦))
5853, 50, 57syl2anc 696 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑦 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑦) = 𝑦))
5949, 58mpbid 222 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹𝑦) = 𝑦)
6056, 59uneq12d 3911 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → ((𝐹𝑥) ∪ (𝐹𝑦)) = (𝑥𝑦))
6152, 60eqtrd 2794 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥𝑦)) = (𝑥𝑦))
6248, 51unssd 3932 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥𝑦) ⊆ 𝐵)
63 vex 3343 . . . . . . . . . 10 𝑥 ∈ V
64 vex 3343 . . . . . . . . . 10 𝑦 ∈ V
6563, 64unex 7121 . . . . . . . . 9 (𝑥𝑦) ∈ V
6665elpw 4308 . . . . . . . 8 ((𝑥𝑦) ∈ 𝒫 𝐵 ↔ (𝑥𝑦) ⊆ 𝐵)
6762, 66sylibr 224 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥𝑦) ∈ 𝒫 𝐵)
68 fnelfp 6605 . . . . . . 7 ((𝐹 Fn 𝒫 𝐵 ∧ (𝑥𝑦) ∈ 𝒫 𝐵) → ((𝑥𝑦) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝑥𝑦)) = (𝑥𝑦)))
6953, 67, 68syl2anc 696 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → ((𝑥𝑦) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝑥𝑦)) = (𝑥𝑦)))
7061, 69mpbird 247 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥𝑦) ∈ dom (𝐹 ∩ I ))
71 eqid 2760 . . . . 5 {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )}
7232, 37, 70, 71mretopd 21098 . . . 4 (𝜑 → ({𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOn‘𝐵) ∧ dom (𝐹 ∩ I ) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )})))
7372simpld 477 . . 3 (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOn‘𝐵))
7416, 73eqeltrd 2839 . 2 (𝜑𝐽 ∈ (TopOn‘𝐵))
75 topontop 20920 . . . . . 6 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
7674, 75syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
77 eqid 2760 . . . . . 6 (mrCls‘(Clsd‘𝐽)) = (mrCls‘(Clsd‘𝐽))
7877mrccls 21085 . . . . 5 (𝐽 ∈ Top → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
7976, 78syl 17 . . . 4 (𝜑 → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
8072simprd 482 . . . . . 6 (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )}))
8116fveq2d 6356 . . . . . 6 (𝜑 → (Clsd‘𝐽) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )}))
8280, 81eqtr4d 2797 . . . . 5 (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘𝐽))
8382fveq2d 6356 . . . 4 (𝜑 → (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘(Clsd‘𝐽)))
8479, 83eqtr4d 2797 . . 3 (𝜑 → (cls‘𝐽) = (mrCls‘dom (𝐹 ∩ I )))
857, 2, 17, 30, 31ismrcd2 37764 . . 3 (𝜑𝐹 = (mrCls‘dom (𝐹 ∩ I )))
8684, 85eqtr4d 2797 . 2 (𝜑 → (cls‘𝐽) = 𝐹)
8774, 86jca 555 1 (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  {crab 3054  cdif 3712  cun 3713  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302   I cid 5173  dom cdm 5266   Fn wfn 6044  wf 6045  cfv 6049  mrClscmrc 16445  Topctop 20900  TopOnctopon 20917  Clsdccld 21022  clsccl 21024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-mre 16448  df-mrc 16449  df-top 20901  df-topon 20918  df-cld 21025  df-cls 21027
This theorem is referenced by: (None)
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