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Theorem kqnrmlem1 22494
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 22478 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
32adantr 484 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
4 topontop 21664 . . 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 22479 . . . . . . 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 22016 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑧 ∈ (KQ‘𝐽)) → (𝐹𝑧) ∈ 𝐽)
118, 9, 10syl2anc 587 . . . . 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 4088 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽)))
14 cnclima 22019 . . . . . 6 ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑤 ∈ (Clsd‘(KQ‘𝐽))) → (𝐹𝑤) ∈ (Clsd‘𝐽))
158, 13, 14syl2anc 587 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (𝐹𝑤) ∈ (Clsd‘𝐽))
1612elin2d 4089 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧)
17 elpwi 4497 . . . . . 6 (𝑤 ∈ 𝒫 𝑧𝑤𝑧)
18 imass2 5939 . . . . . 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 22106 . . . . 5 ((𝐽 ∈ Nrm ∧ ((𝐹𝑧) ∈ 𝐽 ∧ (𝐹𝑤) ∈ (Clsd‘𝐽) ∧ (𝐹𝑤) ⊆ (𝐹𝑧))) → ∃𝑢𝐽 ((𝐹𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))
216, 11, 15, 19, 20syl13anc 1373 . . . 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 22485 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢𝐽) → (𝐹𝑢) ∈ (KQ‘𝐽))
2522, 23, 24syl2anc 587 . . . . 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 22476 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
28 fnfun 6438 . . . . . . . 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 484 . . . . . . . . 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 2738 . . . . . . . . . 10 (KQ‘𝐽) = (KQ‘𝐽)
3231cldss 21780 . . . . . . . . 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 21665 . . . . . . . . 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 3918 . . . . . . 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 6421 . . . . . . 7 ((Fun 𝐹𝑤 ⊆ ran 𝐹) → ((𝐹𝑤) ⊆ 𝑢𝑤 ⊆ (𝐹𝑢)))
3829, 36, 37syl2anc 587 . . . . . 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 21664 . . . . . . . . . 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 4828 . . . . . . . . . 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 2738 . . . . . . . . . 10 𝐽 = 𝐽
4544clscld 21798 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽))
4641, 43, 45syl2anc 587 . . . . . . . 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 22486 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)))
4822, 46, 47syl2anc 587 . . . . . . 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 21807 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢))
5041, 43, 49syl2anc 587 . . . . . . . 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 5939 . . . . . . . 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 21823 . . . . . . 7 (((𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) → ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢)))
5448, 52, 53syl2anc 587 . . . . . 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 21810 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑢 𝐽) → ((cls‘𝐽)‘𝑢) ⊆ 𝐽)
5741, 43, 56syl2anc 587 . . . . . . . . 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 6440 . . . . . . . . . . 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 21665 . . . . . . . . . . 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 2773 . . . . . . . . 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 3918 . . . . . . . 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 6831 . . . . . . . 8 ((Fun 𝐹 ∧ ((cls‘𝐽)‘𝑢) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧 ↔ ((cls‘𝐽)‘𝑢) ⊆ (𝐹𝑧)))
6529, 63, 64syl2anc 587 . . . . . . 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 260 . . . . . 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 3887 . . . . 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 3903 . . . . . . 7 (𝑚 = (𝐹𝑢) → (𝑤𝑚𝑤 ⊆ (𝐹𝑢)))
69 fveq2 6674 . . . . . . . 8 (𝑚 = (𝐹𝑢) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹𝑢)))
7069sseq1d 3908 . . . . . . 7 (𝑚 = (𝐹𝑢) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧 ↔ ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ 𝑧))
7168, 70anbi12d 634 . . . . . 6 (𝑚 = (𝐹𝑢) → ((𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧) ↔ (𝑤 ⊆ (𝐹𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ 𝑧)))
7271rspcev 3526 . . . . 5 (((𝐹𝑢) ∈ (KQ‘𝐽) ∧ (𝑤 ⊆ (𝐹𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹𝑢)) ⊆ 𝑧)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))
7325, 39, 67, 72syl12anc 836 . . . 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 3201 . . 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 3103 . 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 22086 . 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 586 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wral 3053  wrex 3054  {crab 3057  cin 3842  wss 3843  𝒫 cpw 4488   cuni 4796  cmpt 5110  ccnv 5524  dom cdm 5525  ran crn 5526  cima 5528  Fun wfun 6333   Fn wfn 6334  cfv 6339  (class class class)co 7170  Topctop 21644  TopOnctopon 21661  Clsdccld 21767  clsccl 21769   Cn ccn 21975  Nrmcnrm 22061  KQckq 22444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-qtop 16883  df-top 21645  df-topon 21662  df-cld 21770  df-cls 21772  df-cn 21978  df-nrm 22068  df-kq 22445
This theorem is referenced by:  kqnrm  22503
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