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Theorem istopclsd 42688
Description: A closure function which satisfies sscls 23080, clsidm 23091, cls0 23104, and clsun 36311 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 6738 . . . . . . . 8 (𝜑𝐹 Fn 𝒫 𝐵)
43adantr 480 . . . . . . 7 ((𝜑𝑧 ∈ 𝒫 𝐵) → 𝐹 Fn 𝒫 𝐵)
5 difss 4146 . . . . . . . . 9 (𝐵𝑧) ⊆ 𝐵
6 istopclsd.b . . . . . . . . . 10 (𝜑𝐵𝑉)
7 elpw2g 5339 . . . . . . . . . 10 (𝐵𝑉 → ((𝐵𝑧) ∈ 𝒫 𝐵 ↔ (𝐵𝑧) ⊆ 𝐵))
86, 7syl 17 . . . . . . . . 9 (𝜑 → ((𝐵𝑧) ∈ 𝒫 𝐵 ↔ (𝐵𝑧) ⊆ 𝐵))
95, 8mpbiri 258 . . . . . . . 8 (𝜑 → (𝐵𝑧) ∈ 𝒫 𝐵)
109adantr 480 . . . . . . 7 ((𝜑𝑧 ∈ 𝒫 𝐵) → (𝐵𝑧) ∈ 𝒫 𝐵)
11 fnelfp 7195 . . . . . . 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 223 . . . . 5 ((𝜑𝑧 ∈ 𝒫 𝐵) → ((𝐹‘(𝐵𝑧)) = (𝐵𝑧) ↔ (𝐵𝑧) ∈ dom (𝐹 ∩ I )))
1413rabbidva 3440 . . . 4 (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐹‘(𝐵𝑧)) = (𝐵𝑧)} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )})
151, 14eqtrid 2787 . . 3 (𝜑𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )})
16 istopclsd.e . . . . . 6 ((𝜑𝑥𝐵) → 𝑥 ⊆ (𝐹𝑥))
17 simp1 1135 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → 𝜑)
18 simp2 1136 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → 𝑥𝐵)
19 simp3 1137 . . . . . . . . . 10 ((𝜑𝑥𝐵𝑦𝑥) → 𝑦𝑥)
2019, 18sstrd 4006 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → 𝑦𝐵)
21 istopclsd.u . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))
2217, 18, 20, 21syl3anc 1370 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝑥) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))
23 ssequn2 4199 . . . . . . . . . . 11 (𝑦𝑥 ↔ (𝑥𝑦) = 𝑥)
2423biimpi 216 . . . . . . . . . 10 (𝑦𝑥 → (𝑥𝑦) = 𝑥)
25243ad2ant3 1134 . . . . . . . . 9 ((𝜑𝑥𝐵𝑦𝑥) → (𝑥𝑦) = 𝑥)
2625fveq2d 6911 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝑥) → (𝐹‘(𝑥𝑦)) = (𝐹𝑥))
2722, 26eqtr3d 2777 . . . . . . 7 ((𝜑𝑥𝐵𝑦𝑥) → ((𝐹𝑥) ∪ (𝐹𝑦)) = (𝐹𝑥))
28 ssequn2 4199 . . . . . . 7 ((𝐹𝑦) ⊆ (𝐹𝑥) ↔ ((𝐹𝑥) ∪ (𝐹𝑦)) = (𝐹𝑥))
2927, 28sylibr 234 . . . . . 6 ((𝜑𝑥𝐵𝑦𝑥) → (𝐹𝑦) ⊆ (𝐹𝑥))
30 istopclsd.i . . . . . 6 ((𝜑𝑥𝐵) → (𝐹‘(𝐹𝑥)) = (𝐹𝑥))
316, 2, 16, 29, 30ismrcd1 42686 . . . . 5 (𝜑 → dom (𝐹 ∩ I ) ∈ (Moore‘𝐵))
32 istopclsd.z . . . . . 6 (𝜑 → (𝐹‘∅) = ∅)
33 0elpw 5362 . . . . . . 7 ∅ ∈ 𝒫 𝐵
34 fnelfp 7195 . . . . . . 7 ((𝐹 Fn 𝒫 𝐵 ∧ ∅ ∈ 𝒫 𝐵) → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
353, 33, 34sylancl 586 . . . . . 6 (𝜑 → (∅ ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘∅) = ∅))
3632, 35mpbird 257 . . . . 5 (𝜑 → ∅ ∈ dom (𝐹 ∩ I ))
37 simp1 1135 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝜑)
38 inss1 4245 . . . . . . . . . . . . 13 (𝐹 ∩ I ) ⊆ 𝐹
39 dmss 5916 . . . . . . . . . . . . 13 ((𝐹 ∩ I ) ⊆ 𝐹 → dom (𝐹 ∩ I ) ⊆ dom 𝐹)
4038, 39ax-mp 5 . . . . . . . . . . . 12 dom (𝐹 ∩ I ) ⊆ dom 𝐹
4140, 2fssdm 6756 . . . . . . . . . . 11 (𝜑 → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
42413ad2ant1 1132 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → dom (𝐹 ∩ I ) ⊆ 𝒫 𝐵)
43 simp2 1136 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ dom (𝐹 ∩ I ))
4442, 43sseldd 3996 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥 ∈ 𝒫 𝐵)
4544elpwid 4614 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑥𝐵)
46 simp3 1137 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ dom (𝐹 ∩ I ))
4742, 46sseldd 3996 . . . . . . . . 9 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦 ∈ 𝒫 𝐵)
4847elpwid 4614 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝑦𝐵)
4937, 45, 48, 21syl3anc 1370 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥𝑦)) = ((𝐹𝑥) ∪ (𝐹𝑦)))
5033ad2ant1 1132 . . . . . . . . . 10 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → 𝐹 Fn 𝒫 𝐵)
51 fnelfp 7195 . . . . . . . . . 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 232 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹𝑥) = 𝑥)
54 fnelfp 7195 . . . . . . . . . 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 232 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹𝑦) = 𝑦)
5753, 56uneq12d 4179 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → ((𝐹𝑥) ∪ (𝐹𝑦)) = (𝑥𝑦))
5849, 57eqtrd 2775 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝐹‘(𝑥𝑦)) = (𝑥𝑦))
5945, 48unssd 4202 . . . . . . . 8 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥𝑦) ⊆ 𝐵)
60 vex 3482 . . . . . . . . . 10 𝑥 ∈ V
61 vex 3482 . . . . . . . . . 10 𝑦 ∈ V
6260, 61unex 7763 . . . . . . . . 9 (𝑥𝑦) ∈ V
6362elpw 4609 . . . . . . . 8 ((𝑥𝑦) ∈ 𝒫 𝐵 ↔ (𝑥𝑦) ⊆ 𝐵)
6459, 63sylibr 234 . . . . . . 7 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥𝑦) ∈ 𝒫 𝐵)
65 fnelfp 7195 . . . . . . 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 257 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 ∩ I ) ∧ 𝑦 ∈ dom (𝐹 ∩ I )) → (𝑥𝑦) ∈ dom (𝐹 ∩ I ))
68 eqid 2735 . . . . 5 {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )} = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )}
6931, 36, 67, 68mretopd 23116 . . . 4 (𝜑 → ({𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOn‘𝐵) ∧ dom (𝐹 ∩ I ) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )})))
7069simpld 494 . . 3 (𝜑 → {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )} ∈ (TopOn‘𝐵))
7115, 70eqeltrd 2839 . 2 (𝜑𝐽 ∈ (TopOn‘𝐵))
72 topontop 22935 . . . . . 6 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
7371, 72syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
74 eqid 2735 . . . . . 6 (mrCls‘(Clsd‘𝐽)) = (mrCls‘(Clsd‘𝐽))
7574mrccls 23103 . . . . 5 (𝐽 ∈ Top → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
7673, 75syl 17 . . . 4 (𝜑 → (cls‘𝐽) = (mrCls‘(Clsd‘𝐽)))
7769simprd 495 . . . . . 6 (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )}))
7815fveq2d 6911 . . . . . 6 (𝜑 → (Clsd‘𝐽) = (Clsd‘{𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ dom (𝐹 ∩ I )}))
7977, 78eqtr4d 2778 . . . . 5 (𝜑 → dom (𝐹 ∩ I ) = (Clsd‘𝐽))
8079fveq2d 6911 . . . 4 (𝜑 → (mrCls‘dom (𝐹 ∩ I )) = (mrCls‘(Clsd‘𝐽)))
8176, 80eqtr4d 2778 . . 3 (𝜑 → (cls‘𝐽) = (mrCls‘dom (𝐹 ∩ I )))
826, 2, 16, 29, 30ismrcd2 42687 . . 3 (𝜑𝐹 = (mrCls‘dom (𝐹 ∩ I )))
8381, 82eqtr4d 2778 . 2 (𝜑 → (cls‘𝐽) = 𝐹)
8471, 83jca 511 1 (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ (cls‘𝐽) = 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   I cid 5582  dom cdm 5689   Fn wfn 6558  wf 6559  cfv 6563  mrClscmrc 17628  Topctop 22915  TopOnctopon 22932  Clsdccld 23040  clsccl 23042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-mre 17631  df-mrc 17632  df-top 22916  df-topon 22933  df-cld 23043  df-cls 23045
This theorem is referenced by: (None)
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