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Theorem kqnrmlem2 23692
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 22859 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21adantr 479 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Top)
3 simplr 767 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (KQ‘𝐽) ∈ Nrm)
4 simpll 765 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝐽 ∈ (TopOn‘𝑋))
5 simprl 769 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑧𝐽)
6 kqval.2 . . . . . . 7 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
76kqopn 23682 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹𝑧) ∈ (KQ‘𝐽))
84, 5, 7syl2anc 582 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹𝑧) ∈ (KQ‘𝐽))
9 simprr 771 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))
109elin1d 4196 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘𝐽))
116kqcld 23683 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ (Clsd‘𝐽)) → (𝐹𝑤) ∈ (Clsd‘(KQ‘𝐽)))
124, 10, 11syl2anc 582 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹𝑤) ∈ (Clsd‘(KQ‘𝐽)))
139elin2d 4197 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
14 elpwi 4611 . . . . . 6 (𝑤 ∈ 𝒫 𝑧𝑤𝑧)
15 imass2 6107 . . . . . 6 (𝑤𝑧 → (𝐹𝑤) ⊆ (𝐹𝑧))
1613, 14, 153syl 18 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹𝑤) ⊆ (𝐹𝑧))
17 nrmsep3 23303 . . . . 5 (((KQ‘𝐽) ∈ Nrm ∧ ((𝐹𝑧) ∈ (KQ‘𝐽) ∧ (𝐹𝑤) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹𝑤) ⊆ (𝐹𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))
183, 8, 12, 16, 17syl13anc 1369 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))
19 simplll 773 . . . . . . 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))
206kqid 23676 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
2119, 20syl 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)))
22 simprl 769 . . . . . 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))
23 cnima 23213 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑚 ∈ (KQ‘𝐽)) → (𝐹𝑚) ∈ 𝐽)
2421, 22, 23syl2anc 582 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → (𝐹𝑚) ∈ 𝐽)
25 simprrl 779 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → (𝐹𝑤) ⊆ 𝑚)
266kqffn 23673 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27 fnfun 6655 . . . . . . . 8 (𝐹 Fn 𝑋 → Fun 𝐹)
2819, 26, 273syl 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)
2910adantr 479 . . . . . . . . 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))
30 eqid 2725 . . . . . . . . . 10 𝐽 = 𝐽
3130cldss 22977 . . . . . . . . 9 (𝑤 ∈ (Clsd‘𝐽) → 𝑤 𝐽)
3229, 31syl 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)
33 fndm 6658 . . . . . . . . . 10 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
3419, 26, 333syl 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)
35 toponuni 22860 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3619, 35syl 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)
3734, 36eqtrd 2765 . . . . . . . 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)
3832, 37sseqtrrd 4018 . . . . . . 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)
39 funimass3 7062 . . . . . . 7 ((Fun 𝐹𝑤 ⊆ dom 𝐹) → ((𝐹𝑤) ⊆ 𝑚𝑤 ⊆ (𝐹𝑚)))
4028, 38, 39syl2anc 582 . . . . . 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)))
4125, 40mpbid 231 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑤 ⊆ (𝐹𝑚))
426kqtopon 23675 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
43 topontop 22859 . . . . . . . . . 10 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
4419, 42, 433syl 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)
45 elssuni 4941 . . . . . . . . . 10 (𝑚 ∈ (KQ‘𝐽) → 𝑚 (KQ‘𝐽))
4645ad2antrl 726 . . . . . . . . 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))
47 eqid 2725 . . . . . . . . . 10 (KQ‘𝐽) = (KQ‘𝐽)
4847clscld 22995 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑚 (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽)))
4944, 46, 48syl2anc 582 . . . . . . . 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)))
50 cnclima 23216 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽))
5121, 49, 50syl2anc 582 . . . . . . 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))
5247sscls 23004 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑚 (KQ‘𝐽)) → 𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚))
5344, 46, 52syl2anc 582 . . . . . . . 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))
54 imass2 6107 . . . . . . . 8 (𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚) → (𝐹𝑚) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
5553, 54syl 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)))
5630clsss2 23020 . . . . . . 7 (((𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽) ∧ (𝐹𝑚) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚))) → ((cls‘𝐽)‘(𝐹𝑚)) ⊆ (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)))
5751, 55, 56syl2anc 582 . . . . . 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)))
58 simprrr 780 . . . . . . . 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))
59 imass2 6107 . . . . . . . 8 (((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ⊆ (𝐹 “ (𝐹𝑧)))
6058, 59syl 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)))
615adantr 479 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑧𝐽)
626kqsat 23679 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹 “ (𝐹𝑧)) = 𝑧)
6319, 61, 62syl2anc 582 . . . . . . 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)
6460, 63sseqtrd 4017 . . . . . 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)
6557, 64sstrd 3987 . . . . 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)
66 sseq2 4003 . . . . . . 7 (𝑢 = (𝐹𝑚) → (𝑤𝑢𝑤 ⊆ (𝐹𝑚)))
67 fveq2 6896 . . . . . . . 8 (𝑢 = (𝐹𝑚) → ((cls‘𝐽)‘𝑢) = ((cls‘𝐽)‘(𝐹𝑚)))
6867sseq1d 4008 . . . . . . 7 (𝑢 = (𝐹𝑚) → (((cls‘𝐽)‘𝑢) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(𝐹𝑚)) ⊆ 𝑧))
6966, 68anbi12d 630 . . . . . 6 (𝑢 = (𝐹𝑚) → ((𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧) ↔ (𝑤 ⊆ (𝐹𝑚) ∧ ((cls‘𝐽)‘(𝐹𝑚)) ⊆ 𝑧)))
7069rspcev 3606 . . . . 5 (((𝐹𝑚) ∈ 𝐽 ∧ (𝑤 ⊆ (𝐹𝑚) ∧ ((cls‘𝐽)‘(𝐹𝑚)) ⊆ 𝑧)) → ∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
7124, 41, 65, 70syl12anc 835 . . . 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))
7218, 71rexlimddv 3150 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → ∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
7372ralrimivva 3190 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → ∀𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
74 isnrm 23283 . 2 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧)))
752, 73, 74sylanbrc 581 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  {crab 3418  cin 3943  wss 3944  𝒫 cpw 4604   cuni 4909  cmpt 5232  ccnv 5677  dom cdm 5678  ran crn 5679  cima 5681  Fun wfun 6543   Fn wfn 6544  cfv 6549  (class class class)co 7419  Topctop 22839  TopOnctopon 22856  Clsdccld 22964  clsccl 22966   Cn ccn 23172  Nrmcnrm 23258  KQckq 23641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-qtop 17492  df-top 22840  df-topon 22857  df-cld 22967  df-cls 22969  df-cn 23175  df-nrm 23265  df-kq 23642
This theorem is referenced by:  kqnrm  23700
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