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Theorem kqnrmlem2 21749
Description: If the Kolmogorov quotient of a space is normal then so is the original space. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqnrmlem2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqnrmlem2
Dummy variables 𝑚 𝑤 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 20920 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21adantr 472 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Top)
3 simplr 809 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (KQ‘𝐽) ∈ Nrm)
4 simpll 807 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝐽 ∈ (TopOn‘𝑋))
5 simprl 811 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑧𝐽)
6 kqval.2 . . . . . . 7 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
76kqopn 21739 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹𝑧) ∈ (KQ‘𝐽))
84, 5, 7syl2anc 696 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹𝑧) ∈ (KQ‘𝐽))
9 inss1 3976 . . . . . . 7 ((Clsd‘𝐽) ∩ 𝒫 𝑧) ⊆ (Clsd‘𝐽)
10 simprr 813 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))
119, 10sseldi 3742 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘𝐽))
126kqcld 21740 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ (Clsd‘𝐽)) → (𝐹𝑤) ∈ (Clsd‘(KQ‘𝐽)))
134, 11, 12syl2anc 696 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹𝑤) ∈ (Clsd‘(KQ‘𝐽)))
14 inss2 3977 . . . . . . 7 ((Clsd‘𝐽) ∩ 𝒫 𝑧) ⊆ 𝒫 𝑧
1514, 10sseldi 3742 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
16 elpwi 4312 . . . . . 6 (𝑤 ∈ 𝒫 𝑧𝑤𝑧)
17 imass2 5659 . . . . . 6 (𝑤𝑧 → (𝐹𝑤) ⊆ (𝐹𝑧))
1815, 16, 173syl 18 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹𝑤) ⊆ (𝐹𝑧))
19 nrmsep3 21361 . . . . 5 (((KQ‘𝐽) ∈ Nrm ∧ ((𝐹𝑧) ∈ (KQ‘𝐽) ∧ (𝐹𝑤) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹𝑤) ⊆ (𝐹𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))
203, 8, 13, 18, 19syl13anc 1479 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))
21 simplll 815 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝐽 ∈ (TopOn‘𝑋))
226kqid 21733 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
2321, 22syl 17 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
24 simprl 811 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑚 ∈ (KQ‘𝐽))
25 cnima 21271 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑚 ∈ (KQ‘𝐽)) → (𝐹𝑚) ∈ 𝐽)
2623, 24, 25syl2anc 696 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → (𝐹𝑚) ∈ 𝐽)
27 simprrl 823 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → (𝐹𝑤) ⊆ 𝑚)
286kqffn 21730 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
29 fnfun 6149 . . . . . . . 8 (𝐹 Fn 𝑋 → Fun 𝐹)
3021, 28, 293syl 18 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → Fun 𝐹)
3111adantr 472 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑤 ∈ (Clsd‘𝐽))
32 eqid 2760 . . . . . . . . . 10 𝐽 = 𝐽
3332cldss 21035 . . . . . . . . 9 (𝑤 ∈ (Clsd‘𝐽) → 𝑤 𝐽)
3431, 33syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑤 𝐽)
35 fndm 6151 . . . . . . . . . 10 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
3621, 28, 353syl 18 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → dom 𝐹 = 𝑋)
37 toponuni 20921 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3821, 37syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑋 = 𝐽)
3936, 38eqtrd 2794 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → dom 𝐹 = 𝐽)
4034, 39sseqtr4d 3783 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑤 ⊆ dom 𝐹)
41 funimass3 6496 . . . . . . 7 ((Fun 𝐹𝑤 ⊆ dom 𝐹) → ((𝐹𝑤) ⊆ 𝑚𝑤 ⊆ (𝐹𝑚)))
4230, 40, 41syl2anc 696 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → ((𝐹𝑤) ⊆ 𝑚𝑤 ⊆ (𝐹𝑚)))
4327, 42mpbid 222 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑤 ⊆ (𝐹𝑚))
446kqtopon 21732 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
45 topontop 20920 . . . . . . . . . 10 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
4621, 44, 453syl 18 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → (KQ‘𝐽) ∈ Top)
47 elssuni 4619 . . . . . . . . . 10 (𝑚 ∈ (KQ‘𝐽) → 𝑚 (KQ‘𝐽))
4847ad2antrl 766 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑚 (KQ‘𝐽))
49 eqid 2760 . . . . . . . . . 10 (KQ‘𝐽) = (KQ‘𝐽)
5049clscld 21053 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑚 (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽)))
5146, 48, 50syl2anc 696 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽)))
52 cnclima 21274 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽))
5323, 51, 52syl2anc 696 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽))
5449sscls 21062 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑚 (KQ‘𝐽)) → 𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚))
5546, 48, 54syl2anc 696 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚))
56 imass2 5659 . . . . . . . 8 (𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚) → (𝐹𝑚) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
5755, 56syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → (𝐹𝑚) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
5832clsss2 21078 . . . . . . 7 (((𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽) ∧ (𝐹𝑚) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚))) → ((cls‘𝐽)‘(𝐹𝑚)) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
5953, 57, 58syl2anc 696 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → ((cls‘𝐽)‘(𝐹𝑚)) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
60 simprrr 824 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧))
61 imass2 5659 . . . . . . . 8 (((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ (𝐹 “ (𝐹𝑧)))
6260, 61syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ (𝐹 “ (𝐹𝑧)))
635adantr 472 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑧𝐽)
646kqsat 21736 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹 “ (𝐹𝑧)) = 𝑧)
6521, 63, 64syl2anc 696 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → (𝐹 “ (𝐹𝑧)) = 𝑧)
6662, 65sseqtrd 3782 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ 𝑧)
6759, 66sstrd 3754 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → ((cls‘𝐽)‘(𝐹𝑚)) ⊆ 𝑧)
68 sseq2 3768 . . . . . . 7 (𝑢 = (𝐹𝑚) → (𝑤𝑢𝑤 ⊆ (𝐹𝑚)))
69 fveq2 6352 . . . . . . . 8 (𝑢 = (𝐹𝑚) → ((cls‘𝐽)‘𝑢) = ((cls‘𝐽)‘(𝐹𝑚)))
7069sseq1d 3773 . . . . . . 7 (𝑢 = (𝐹𝑚) → (((cls‘𝐽)‘𝑢) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(𝐹𝑚)) ⊆ 𝑧))
7168, 70anbi12d 749 . . . . . 6 (𝑢 = (𝐹𝑚) → ((𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧) ↔ (𝑤 ⊆ (𝐹𝑚) ∧ ((cls‘𝐽)‘(𝐹𝑚)) ⊆ 𝑧)))
7271rspcev 3449 . . . . 5 (((𝐹𝑚) ∈ 𝐽 ∧ (𝑤 ⊆ (𝐹𝑚) ∧ ((cls‘𝐽)‘(𝐹𝑚)) ⊆ 𝑧)) → ∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
7326, 43, 67, 72syl12anc 1475 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → ∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
7420, 73rexlimddv 3173 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → ∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
7574ralrimivva 3109 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → ∀𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
76 isnrm 21341 . 2 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧)))
772, 75, 76sylanbrc 701 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  cin 3714  wss 3715  𝒫 cpw 4302   cuni 4588  cmpt 4881  ccnv 5265  dom cdm 5266  ran crn 5267  cima 5269  Fun wfun 6043   Fn wfn 6044  cfv 6049  (class class class)co 6813  Topctop 20900  TopOnctopon 20917  Clsdccld 21022  clsccl 21024   Cn ccn 21230  Nrmcnrm 21316  KQckq 21698
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-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-qtop 16369  df-top 20901  df-topon 20918  df-cld 21025  df-cls 21027  df-cn 21233  df-nrm 21323  df-kq 21699
This theorem is referenced by:  kqnrm  21757
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