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Theorem kqnrmlem1 21767
Description: A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
kqnrmlem1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem kqnrmlem1
Dummy variables 𝑚 𝑤 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . 5 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqtopon 21751 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
32adantr 466 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
4 topontop 20938 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
53, 4syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Top)
6 simplr 752 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐽 ∈ Nrm)
71kqid 21752 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
87ad2antrr 705 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
9 simprl 754 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑧 ∈ (KQ‘𝐽))
10 cnima 21290 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑧 ∈ (KQ‘𝐽)) → (𝐹𝑧) ∈ 𝐽)
118, 9, 10syl2anc 573 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (𝐹𝑧) ∈ 𝐽)
12 inss1 3981 . . . . . . 7 ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧) ⊆ (Clsd‘(KQ‘𝐽))
13 simprr 756 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))
1412, 13sseldi 3750 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽)))
15 cnclima 21293 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑤 ∈ (Clsd‘(KQ‘𝐽))) → (𝐹𝑤) ∈ (Clsd‘𝐽))
168, 14, 15syl2anc 573 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (𝐹𝑤) ∈ (Clsd‘𝐽))
17 inss2 3982 . . . . . . 7 ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧) ⊆ 𝒫 𝑧
1817, 13sseldi 3750 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
19 elpwi 4307 . . . . . 6 (𝑤 ∈ 𝒫 𝑧𝑤𝑧)
20 imass2 5642 . . . . . 6 (𝑤𝑧 → (𝐹𝑤) ⊆ (𝐹𝑧))
2118, 19, 203syl 18 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (𝐹𝑤) ⊆ (𝐹𝑧))
22 nrmsep3 21380 . . . . 5 ((𝐽 ∈ Nrm ∧ ((𝐹𝑧) ∈ 𝐽 ∧ (𝐹𝑤) ∈ (Clsd‘𝐽) ∧ (𝐹𝑤) ⊆ (𝐹𝑧))) → ∃𝑢𝐽 ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))
236, 11, 16, 21, 22syl13anc 1478 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → ∃𝑢𝐽 ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))
24 simplll 758 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝐽 ∈ (TopOn‘𝑋))
25 simprl 754 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝑢𝐽)
261kqopn 21758 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢𝐽) → (𝐹𝑢) ∈ (KQ‘𝐽))
2724, 25, 26syl2anc 573 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → (𝐹𝑢) ∈ (KQ‘𝐽))
28 simprrl 766 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → (𝐹𝑤) ⊆ 𝑢)
291kqffn 21749 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
30 fnfun 6128 . . . . . . . 8 (𝐹 Fn 𝑋 → Fun 𝐹)
3124, 29, 303syl 18 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → Fun 𝐹)
3214adantr 466 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽)))
33 eqid 2771 . . . . . . . . . 10 (KQ‘𝐽) = (KQ‘𝐽)
3433cldss 21054 . . . . . . . . 9 (𝑤 ∈ (Clsd‘(KQ‘𝐽)) → 𝑤 (KQ‘𝐽))
3532, 34syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝑤 (KQ‘𝐽))
36 toponuni 20939 . . . . . . . . 9 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = (KQ‘𝐽))
3724, 2, 363syl 18 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → ran 𝐹 = (KQ‘𝐽))
3835, 37sseqtr4d 3791 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝑤 ⊆ ran 𝐹)
39 funimass1 6111 . . . . . . 7 ((Fun 𝐹𝑤 ⊆ ran 𝐹) → ((𝐹𝑤) ⊆ 𝑢𝑤 ⊆ (𝐹𝑢)))
4031, 38, 39syl2anc 573 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → ((𝐹𝑤) ⊆ 𝑢𝑤 ⊆ (𝐹𝑢)))
4128, 40mpd 15 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝑤 ⊆ (𝐹𝑢))
42 topontop 20938 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4324, 42syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝐽 ∈ Top)
44 elssuni 4603 . . . . . . . . . 10 (𝑢𝐽𝑢 𝐽)
4544ad2antrl 707 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝑢 𝐽)
46 eqid 2771 . . . . . . . . . 10 𝐽 = 𝐽
4746clscld 21072 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
4843, 45, 47syl2anc 573 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
491kqcld 21759 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)))
5024, 48, 49syl2anc 573 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)))
5146sscls 21081 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
5243, 45, 51syl2anc 573 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
53 imass2 5642 . . . . . . . 8 (𝑢 ⊆ ((cls‘𝐽)‘𝑢) → (𝐹𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
5452, 53syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → (𝐹𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
5533clsss2 21097 . . . . . . 7 (((𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
5650, 54, 55syl2anc 573 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
57 simprrr 767 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧))
5846clsss3 21084 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
5943, 45, 58syl2anc 573 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
60 fndm 6130 . . . . . . . . . . 11 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
6124, 29, 603syl 18 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → dom 𝐹 = 𝑋)
62 toponuni 20939 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
6324, 62syl 17 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝑋 = 𝐽)
6461, 63eqtrd 2805 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → dom 𝐹 = 𝐽)
6559, 64sseqtr4d 3791 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ dom 𝐹)
66 funimass3 6476 . . . . . . . 8 ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑢) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧 ↔ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))
6731, 65, 66syl2anc 573 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → ((𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧 ↔ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))
6857, 67mpbird 247 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧)
6956, 68sstrd 3762 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ 𝑧)
70 sseq2 3776 . . . . . . 7 (𝑚 = (𝐹𝑢) → (𝑤𝑚𝑤 ⊆ (𝐹𝑢)))
71 fveq2 6332 . . . . . . . 8 (𝑚 = (𝐹𝑢) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹𝑢)))
7271sseq1d 3781 . . . . . . 7 (𝑚 = (𝐹𝑢) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧 ↔ ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ 𝑧))
7370, 72anbi12d 616 . . . . . 6 (𝑚 = (𝐹𝑢) → ((𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧) ↔ (𝑤 ⊆ (𝐹𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ 𝑧)))
7473rspcev 3460 . . . . 5 (((𝐹𝑢) ∈ (KQ‘𝐽) ∧ (𝑤 ⊆ (𝐹𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ 𝑧)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
7527, 41, 69, 74syl12anc 1474 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
7623, 75rexlimddv 3183 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
7776ralrimivva 3120 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → ∀𝑧 ∈ (KQ‘𝐽)∀𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧)∃𝑚 ∈ (KQ‘𝐽)(𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
78 isnrm 21360 . 2 ((KQ‘𝐽) ∈ Nrm ↔ ((KQ‘𝐽) ∈ Top ∧ ∀𝑧 ∈ (KQ‘𝐽)∀𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧)∃𝑚 ∈ (KQ‘𝐽)(𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧)))
795, 77, 78sylanbrc 572 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  {crab 3065  cin 3722  wss 3723  𝒫 cpw 4297   cuni 4574  cmpt 4863  ccnv 5248  dom cdm 5249  ran crn 5250  cima 5252  Fun wfun 6025   Fn wfn 6026  cfv 6031  (class class class)co 6793  Topctop 20918  TopOnctopon 20935  Clsdccld 21041  clsccl 21043   Cn ccn 21249  Nrmcnrm 21335  KQckq 21717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-qtop 16375  df-top 20919  df-topon 20936  df-cld 21044  df-cls 21046  df-cn 21252  df-nrm 21342  df-kq 21718
This theorem is referenced by:  kqnrm  21776
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