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Theorem istopclsd 39175
Description: A closure function which satisfies sscls 21592, clsidm 21603, cls0 21616, and clsun 33573 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)
32ffnd 6508 . . . . . . . 8 (𝜑𝐹 Fn 𝒫 𝐵)
43adantr 481 . . . . . . 7 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
5 difss 4105 . . . . . . . . 9 (𝐵𝑧) ⊆ 𝐵
6 istopclsd.b . . . . . . . . . 10 (𝜑𝐵𝑉)
7 elpw2g 5238 . . . . . . . . . 10 (𝐵𝑉 → ((𝐵𝑧) ∈ 𝒫 𝐵 ↔ (𝐵𝑧) ⊆ 𝐵))
86, 7syl 17 . . . . . . . . 9 (𝜑 → ((𝐵𝑧) ∈ 𝒫 𝐵 ↔ (𝐵𝑧) ⊆ 𝐵))
95, 8mpbiri 259 . . . . . . . 8 (𝜑 → (𝐵𝑧) ∈ 𝒫 𝐵)
109adantr 481 . . . . . . 7 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐵𝑧) ∈ 𝒫 𝐵)
11 fnelfp 6929 . . . . . . 7 ((𝐹 Fn 𝒫 𝐵 ∧ (𝐵𝑧) ∈ 𝒫 𝐵) → ((𝐵𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)))
124, 10, 11syl2anc 584 . . . . . 6 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((𝐵𝑧) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)))
1312bicomd 224 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((𝐹‘(𝐵𝑧)) = (𝐵𝑧) ↔ (𝐵𝑧) ∈ dom (𝐹 ∩ I )))
1413rabbidva 3476 . . . 4 (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )})
151, 14syl5eq 2865 . . 3 (𝜑𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )})
16 istopclsd.e . . . . . 6 ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))
17 simp1 1128 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → 𝜑)
18 simp2 1129 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → 𝑥𝐵)
19 simp3 1130 . . . . . . . . . 10 ((𝜑𝑥𝐵𝑦𝑥) → 𝑦𝑥)
2019, 18sstrd 3974 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → 𝑦𝐵)
21 istopclsd.u . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))
2217, 18, 20, 21syl3anc 1363 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝑥) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))
23 ssequn2 4156 . . . . . . . . . . 11 (𝑦𝑥 ↔ (𝑥𝑦) = 𝑥)
2423biimpi 217 . . . . . . . . . 10 (𝑦𝑥 → (𝑥𝑦) = 𝑥)
25243ad2ant3 1127 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → (𝑥𝑦) = 𝑥)
2625fveq2d 6667 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝑥) → (𝐹‘(𝑥𝑦)) = (𝐹𝑥))
2722, 26eqtr3d 2855 . . . . . . 7 ((𝜑𝑥𝐵𝑦𝑥) → ((𝐹𝑥) ∪ (𝐹𝑦)) = (𝐹𝑥))
28 ssequn2 4156 . . . . . . 7 ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ ((𝐹𝑥) ∪ (𝐹𝑦)) = (𝐹𝑥))
2927, 28sylibr 235 . . . . . 6 ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))
30 istopclsd.i . . . . . 6 ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
316, 2, 16, 29, 30ismrcd1 39173 . . . . 5 (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
32 istopclsd.z . . . . . 6 (𝜑 → (𝐹‘∅) = ∅)
33 0elpw 5247 . . . . . . 7 ∅ ∈ 𝒫 𝐵
34 fnelfp 6929 . . . . . . 7 ((𝐹 Fn 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
353, 33, 34sylancl 586 . . . . . 6 (𝜑 → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
3632, 35mpbird 258 . . . . 5 (𝜑 → ∅ ∈ dom (𝐹 ∩ I ))
37 simp1 1128 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝜑)
38 inss1 4202 . . . . . . . . . . . . 13 (𝐹 ∩ I ) ⊆ 𝐹
39 dmss 5764 . . . . . . . . . . . . 13 ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
4038, 39ax-mp 5 . . . . . . . . . . . 12 dom (𝐹 ∩ I ) ⊆ dom 𝐹
4140, 2fssdm 6523 . . . . . . . . . . 11 (𝜑 → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
42413ad2ant1 1125 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
43 simp2 1129 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ dom (𝐹 ∩ I ))
4442, 43sseldd 3965 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ 𝒫 𝐵)
4544elpwid 4549 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥𝐵)
46 simp3 1130 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ dom (𝐹 ∩ I ))
4742, 46sseldd 3965 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ 𝒫 𝐵)
4847elpwid 4549 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦𝐵)
4937, 45, 48, 21syl3anc 1363 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))
5033ad2ant1 1125 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝐹 Fn 𝒫 𝐵)
51 fnelfp 6929 . . . . . . . . . 10 ((𝐹 Fn 𝒫 𝐵𝑥 ∈ 𝒫 𝐵) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑥) = 𝑥))
5250, 44, 51syl2anc 584 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑥) = 𝑥))
5343, 52mpbid 233 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹𝑥) = 𝑥)
54 fnelfp 6929 . . . . . . . . . 10 ((𝐹 Fn 𝒫 𝐵𝑦 ∈ 𝒫 𝐵) → (𝑦 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑦) = 𝑦))
5550, 47, 54syl2anc 584 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑦 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑦) = 𝑦))
5646, 55mpbid 233 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹𝑦) = 𝑦)
5753, 56uneq12d 4137 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → ((𝐹𝑥) ∪ (𝐹𝑦)) = (𝑥𝑦))
5849, 57eqtrd 2853 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥𝑦)) = (𝑥𝑦))
5945, 48unssd 4159 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥𝑦) ⊆ 𝐵)
60 vex 3495 . . . . . . . . . 10 𝑥 ∈ V
61 vex 3495 . . . . . . . . . 10 𝑦 ∈ V
6260, 61unex 7458 . . . . . . . . 9 (𝑥𝑦) ∈ V
6362elpw 4542 . . . . . . . 8 ((𝑥𝑦) ∈ 𝒫 𝐵 ↔ (𝑥𝑦) ⊆ 𝐵)
6459, 63sylibr 235 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥𝑦) ∈ 𝒫 𝐵)
65 fnelfp 6929 . . . . . . 7 ((𝐹 Fn 𝒫 𝐵 ∧ (𝑥𝑦) ∈ 𝒫 𝐵) → ((𝑥𝑦) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝑥𝑦)) = (𝑥𝑦)))
6650, 64, 65syl2anc 584 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → ((𝑥𝑦) ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘(𝑥𝑦)) = (𝑥𝑦)))
6758, 66mpbird 258 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥𝑦) ∈ dom (𝐹 ∩ I ))
68 eqid 2818 . . . . 5 {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )}
6931, 36, 67, 68mretopd 21628 . . . 4 (𝜑 → ({𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOn‘𝐵) ∧ dom (𝐹 ∩ I ) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )})))
7069simpld 495 . . 3 (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOn‘𝐵))
7115, 70eqeltrd 2910 . 2 (𝜑𝐽 ∈ (TopOn‘𝐵))
72 topontop 21449 . . . . . 6 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
7371, 72syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
74 eqid 2818 . . . . . 6 (mrCls‘(Clsd‘𝐽)) = (mrCls‘(Clsd‘𝐽))
7574mrccls 21615 . . . . 5 (𝐽 ∈ Top → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
7673, 75syl 17 . . . 4 (𝜑 → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
7769simprd 496 . . . . . 6 (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )}))
7815fveq2d 6667 . . . . . 6 (𝜑 → (Clsd‘𝐽) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )}))
7977, 78eqtr4d 2856 . . . . 5 (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘𝐽))
8079fveq2d 6667 . . . 4 (𝜑 → (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘(Clsd‘𝐽)))
8176, 80eqtr4d 2856 . . 3 (𝜑 → (cls‘𝐽) = (mrCls‘dom (𝐹 ∩ I )))
826, 2, 16, 29, 30ismrcd2 39174 . . 3 (𝜑𝐹 = (mrCls‘dom (𝐹 ∩ I )))
8381, 82eqtr4d 2856 . 2 (𝜑 → (cls‘𝐽) = 𝐹)
8471, 83jca 512 1 (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {crab 3139  cdif 3930  cun 3931  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535   I cid 5452  dom cdm 5548   Fn wfn 6343  wf 6344  cfv 6348  mrClscmrc 16842  Topctop 21429  TopOnctopon 21446  Clsdccld 21552  clsccl 21554
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-rep 5181  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-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  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-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-mre 16845  df-mrc 16846  df-top 21430  df-topon 21447  df-cld 21555  df-cls 21557
This theorem is referenced by: (None)
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