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Theorem kqnrmlem1 23767
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 23751 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
32adantr 480 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
4 topontop 22935 . . 3 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → (KQ‘𝐽) ∈ Top)
53, 4syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Top)
6 simplr 769 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐽 ∈ Nrm)
71kqid 23752 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
87ad2antrr 726 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
9 simprl 771 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑧 ∈ (KQ‘𝐽))
10 cnima 23289 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑧 ∈ (KQ‘𝐽)) → (𝐹𝑧) ∈ 𝐽)
118, 9, 10syl2anc 584 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (𝐹𝑧) ∈ 𝐽)
12 simprr 773 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))
1312elin1d 4214 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽)))
14 cnclima 23292 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑤 ∈ (Clsd‘(KQ‘𝐽))) → (𝐹𝑤) ∈ (Clsd‘𝐽))
158, 13, 14syl2anc 584 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (𝐹𝑤) ∈ (Clsd‘𝐽))
1612elin2d 4215 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
17 elpwi 4612 . . . . . 6 (𝑤 ∈ 𝒫 𝑧𝑤𝑧)
18 imass2 6123 . . . . . 6 (𝑤𝑧 → (𝐹𝑤) ⊆ (𝐹𝑧))
1916, 17, 183syl 18 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (𝐹𝑤) ⊆ (𝐹𝑧))
20 nrmsep3 23379 . . . . 5 ((𝐽 ∈ Nrm ∧ ((𝐹𝑧) ∈ 𝐽 ∧ (𝐹𝑤) ∈ (Clsd‘𝐽) ∧ (𝐹𝑤) ⊆ (𝐹𝑧))) → ∃𝑢𝐽 ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))
216, 11, 15, 19, 20syl13anc 1371 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → ∃𝑢𝐽 ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))
22 simplll 775 . . . . . 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))
23 simprl 771 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝑢𝐽)
241kqopn 23758 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢𝐽) → (𝐹𝑢) ∈ (KQ‘𝐽))
2522, 23, 24syl2anc 584 . . . . 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))
26 simprrl 781 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → (𝐹𝑤) ⊆ 𝑢)
271kqffn 23749 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
28 fnfun 6669 . . . . . . . 8 (𝐹 Fn 𝑋 → Fun 𝐹)
2922, 27, 283syl 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)
3013adantr 480 . . . . . . . . 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)))
31 eqid 2735 . . . . . . . . . 10 (KQ‘𝐽) = (KQ‘𝐽)
3231cldss 23053 . . . . . . . . 9 (𝑤 ∈ (Clsd‘(KQ‘𝐽)) → 𝑤 (KQ‘𝐽))
3330, 32syl 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))
34 toponuni 22936 . . . . . . . . 9 ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = (KQ‘𝐽))
3522, 2, 343syl 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))
3633, 35sseqtrrd 4037 . . . . . . 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)
37 funimass1 6650 . . . . . . 7 ((Fun 𝐹𝑤 ⊆ ran 𝐹) → ((𝐹𝑤) ⊆ 𝑢𝑤 ⊆ (𝐹𝑢)))
3829, 36, 37syl2anc 584 . . . . . 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)))
3926, 38mpd 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))
40 topontop 22935 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4122, 40syl 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)
42 elssuni 4942 . . . . . . . . . 10 (𝑢𝐽𝑢 𝐽)
4342ad2antrl 728 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢𝐽 ∧ ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))) → 𝑢 𝐽)
44 eqid 2735 . . . . . . . . . 10 𝐽 = 𝐽
4544clscld 23071 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
4641, 43, 45syl2anc 584 . . . . . . . 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))
471kqcld 23759 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)))
4822, 46, 47syl2anc 584 . . . . . . 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)))
4944sscls 23080 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
5041, 43, 49syl2anc 584 . . . . . . . 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))
51 imass2 6123 . . . . . . . 8 (𝑢 ⊆ ((cls‘𝐽)‘𝑢) → (𝐹𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
5250, 51syl 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)))
5331clsss2 23096 . . . . . . 7 (((𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
5448, 52, 53syl2anc 584 . . . . . 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)))
55 simprrr 782 . . . . . . 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))
5644clsss3 23083 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
5741, 43, 56syl2anc 584 . . . . . . . . 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)
58 fndm 6672 . . . . . . . . . . 11 (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
5922, 27, 583syl 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)
60 toponuni 22936 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
6122, 60syl 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)
6259, 61eqtrd 2775 . . . . . . . . 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)
6357, 62sseqtrrd 4037 . . . . . . . 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)
64 funimass3 7074 . . . . . . . 8 ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑢) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧 ↔ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))
6529, 63, 64syl2anc 584 . . . . . . 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)))
6655, 65mpbird 257 . . . . . 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)
6754, 66sstrd 4006 . . . . 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)
68 sseq2 4022 . . . . . . 7 (𝑚 = (𝐹𝑢) → (𝑤𝑚𝑤 ⊆ (𝐹𝑢)))
69 fveq2 6907 . . . . . . . 8 (𝑚 = (𝐹𝑢) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹𝑢)))
7069sseq1d 4027 . . . . . . 7 (𝑚 = (𝐹𝑢) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧 ↔ ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ 𝑧))
7168, 70anbi12d 632 . . . . . 6 (𝑚 = (𝐹𝑢) → ((𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧) ↔ (𝑤 ⊆ (𝐹𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ 𝑧)))
7271rspcev 3622 . . . . 5 (((𝐹𝑢) ∈ (KQ‘𝐽) ∧ (𝑤 ⊆ (𝐹𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ 𝑧)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
7325, 39, 67, 72syl12anc 837 . . . 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))
7421, 73rexlimddv 3159 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
7574ralrimivva 3200 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → ∀𝑧 ∈ (KQ‘𝐽)∀𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧)∃𝑚 ∈ (KQ‘𝐽)(𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
76 isnrm 23359 . 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)))
775, 75, 76sylanbrc 583 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912  cmpt 5231  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557   Fn wfn 6558  cfv 6563  (class class class)co 7431  Topctop 22915  TopOnctopon 22932  Clsdccld 23040  clsccl 23042   Cn ccn 23248  Nrmcnrm 23334  KQckq 23717
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-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-qtop 17554  df-top 22916  df-topon 22933  df-cld 23043  df-cls 23045  df-cn 23251  df-nrm 23341  df-kq 23718
This theorem is referenced by:  kqnrm  23776
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