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Theorem kqnrmlem2 22349
 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 21518 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21adantr 484 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Top)
3 simplr 768 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (KQ‘𝐽) ∈ Nrm)
4 simpll 766 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝐽 ∈ (TopOn‘𝑋))
5 simprl 770 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑧𝐽)
6 kqval.2 . . . . . . 7 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
76kqopn 22339 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹𝑧) ∈ (KQ‘𝐽))
84, 5, 7syl2anc 587 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹𝑧) ∈ (KQ‘𝐽))
9 simprr 772 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))
109elin1d 4125 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘𝐽))
116kqcld 22340 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ (Clsd‘𝐽)) → (𝐹𝑤) ∈ (Clsd‘(KQ‘𝐽)))
124, 10, 11syl2anc 587 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹𝑤) ∈ (Clsd‘(KQ‘𝐽)))
139elin2d 4126 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
14 elpwi 4506 . . . . . 6 (𝑤 ∈ 𝒫 𝑧𝑤𝑧)
15 imass2 5932 . . . . . 6 (𝑤𝑧 → (𝐹𝑤) ⊆ (𝐹𝑧))
1613, 14, 153syl 18 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → (𝐹𝑤) ⊆ (𝐹𝑧))
17 nrmsep3 21960 . . . . 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 774 . . . . . . 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 22333 . . . . . . 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 770 . . . . . 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 21870 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑚 ∈ (KQ‘𝐽)) → (𝐹𝑚) ∈ 𝐽)
2421, 22, 23syl2anc 587 . . . . 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 780 . . . . . 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 22330 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
27 fnfun 6423 . . . . . . . 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 484 . . . . . . . . 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 2798 . . . . . . . . . 10 𝐽 = 𝐽
3130cldss 21634 . . . . . . . . 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 6425 . . . . . . . . . 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 21519 . . . . . . . . . 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 2833 . . . . . . . 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 3956 . . . . . . 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 6801 . . . . . . 7 ((Fun 𝐹𝑤 ⊆ dom 𝐹) → ((𝐹𝑤) ⊆ 𝑚𝑤 ⊆ (𝐹𝑚)))
4028, 38, 39syl2anc 587 . . . . . 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 235 . . . . 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 22332 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
43 topontop 21518 . . . . . . . . . 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 4830 . . . . . . . . . 10 (𝑚 ∈ (KQ‘𝐽) → 𝑚 (KQ‘𝐽))
4645ad2antrl 727 . . . . . . . . 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 2798 . . . . . . . . . 10 (KQ‘𝐽) = (KQ‘𝐽)
4847clscld 21652 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑚 (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽)))
4944, 46, 48syl2anc 587 . . . . . . . 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 21873 . . . . . . . 8 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ ((cls‘(KQ‘𝐽))‘𝑚) ∈ (Clsd‘(KQ‘𝐽))) → (𝐹 “ ((cls‘(KQ‘𝐽))‘𝑚)) ∈ (Clsd‘𝐽))
5121, 49, 50syl2anc 587 . . . . . . 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 21661 . . . . . . . . 9 (((KQ‘𝐽) ∈ Top ∧ 𝑚 (KQ‘𝐽)) → 𝑚 ⊆ ((cls‘(KQ‘𝐽))‘𝑚))
5344, 46, 52syl2anc 587 . . . . . . . 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 5932 . . . . . . . 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 21677 . . . . . . 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 587 . . . . . 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 781 . . . . . . . 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 5932 . . . . . . . 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 484 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) ∧ (𝑚 ∈ (KQ‘𝐽) ∧ ((𝐹𝑤) ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ (𝐹𝑧)))) → 𝑧𝐽)
626kqsat 22336 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → (𝐹 “ (𝐹𝑧)) = 𝑧)
6319, 61, 62syl2anc 587 . . . . . . 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 3955 . . . . . 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 3925 . . . . 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 3941 . . . . . . 7 (𝑢 = (𝐹𝑚) → (𝑤𝑢𝑤 ⊆ (𝐹𝑚)))
67 fveq2 6645 . . . . . . . 8 (𝑢 = (𝐹𝑚) → ((cls‘𝐽)‘𝑢) = ((cls‘𝐽)‘(𝐹𝑚)))
6867sseq1d 3946 . . . . . . 7 (𝑢 = (𝐹𝑚) → (((cls‘𝐽)‘𝑢) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(𝐹𝑚)) ⊆ 𝑧))
6966, 68anbi12d 633 . . . . . 6 (𝑢 = (𝐹𝑚) → ((𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧) ↔ (𝑤 ⊆ (𝐹𝑚) ∧ ((cls‘𝐽)‘(𝐹𝑚)) ⊆ 𝑧)))
7069rspcev 3571 . . . . 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 3250 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) ∧ (𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧))) → ∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
7372ralrimivva 3156 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → ∀𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧))
74 isnrm 21940 . 2 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑧𝐽𝑤 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑧)∃𝑢𝐽 (𝑤𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ 𝑧)))
752, 73, 74sylanbrc 586 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Nrm) → 𝐽 ∈ Nrm)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800   ↦ cmpt 5110  ◡ccnv 5518  dom cdm 5519  ran crn 5520   “ cima 5522  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135  Topctop 21498  TopOnctopon 21515  Clsdccld 21621  clsccl 21623   Cn ccn 21829  Nrmcnrm 21915  KQckq 22298 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-qtop 16772  df-top 21499  df-topon 21516  df-cld 21624  df-cls 21626  df-cn 21832  df-nrm 21922  df-kq 22299 This theorem is referenced by:  kqnrm  22357
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