| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | kqval.2 | . . . . 5
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | 
| 2 | 1 | kqtopon 23736 | . . . 4
⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹)) | 
| 3 | 2 | adantr 480 | . . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹)) | 
| 4 |  | topontop 22920 | . . 3
⊢
((KQ‘𝐽) ∈
(TopOn‘ran 𝐹) →
(KQ‘𝐽) ∈
Top) | 
| 5 | 3, 4 | syl 17 | . 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Top) | 
| 6 |  | simplr 768 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐽 ∈ Nrm) | 
| 7 | 1 | kqid 23737 | . . . . . . 7
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽))) | 
| 8 | 7 | ad2antrr 726 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽))) | 
| 9 |  | simprl 770 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑧 ∈ (KQ‘𝐽)) | 
| 10 |  | cnima 23274 | . . . . . 6
⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑧 ∈ (KQ‘𝐽)) → (◡𝐹 “ 𝑧) ∈ 𝐽) | 
| 11 | 8, 9, 10 | syl2anc 584 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (◡𝐹 “ 𝑧) ∈ 𝐽) | 
| 12 |  | simprr 772 | . . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧)) | 
| 13 | 12 | elin1d 4203 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽))) | 
| 14 |  | cnclima 23277 | . . . . . 6
⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑤 ∈ (Clsd‘(KQ‘𝐽))) → (◡𝐹 “ 𝑤) ∈ (Clsd‘𝐽)) | 
| 15 | 8, 13, 14 | syl2anc 584 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (◡𝐹 “ 𝑤) ∈ (Clsd‘𝐽)) | 
| 16 | 12 | elin2d 4204 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧) | 
| 17 |  | elpwi 4606 | . . . . . 6
⊢ (𝑤 ∈ 𝒫 𝑧 → 𝑤 ⊆ 𝑧) | 
| 18 |  | imass2 6119 | . . . . . 6
⊢ (𝑤 ⊆ 𝑧 → (◡𝐹 “ 𝑤) ⊆ (◡𝐹 “ 𝑧)) | 
| 19 | 16, 17, 18 | 3syl 18 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (◡𝐹 “ 𝑤) ⊆ (◡𝐹 “ 𝑧)) | 
| 20 |  | nrmsep3 23364 | . . . . 5
⊢ ((𝐽 ∈ Nrm ∧ ((◡𝐹 “ 𝑧) ∈ 𝐽 ∧ (◡𝐹 “ 𝑤) ∈ (Clsd‘𝐽) ∧ (◡𝐹 “ 𝑤) ⊆ (◡𝐹 “ 𝑧))) → ∃𝑢 ∈ 𝐽 ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧))) | 
| 21 | 6, 11, 15, 19, 20 | syl13anc 1373 | . . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → ∃𝑢 ∈ 𝐽 ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧))) | 
| 22 |  | simplll 774 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 23 |  | simprl 770 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑢 ∈ 𝐽) | 
| 24 | 1 | kqopn 23743 | . . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢 ∈ 𝐽) → (𝐹 “ 𝑢) ∈ (KQ‘𝐽)) | 
| 25 | 22, 23, 24 | syl2anc 584 | . . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → (𝐹 “ 𝑢) ∈ (KQ‘𝐽)) | 
| 26 |  | simprrl 780 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → (◡𝐹 “ 𝑤) ⊆ 𝑢) | 
| 27 | 1 | kqffn 23734 | . . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋) | 
| 28 |  | fnfun 6667 | . . . . . . . 8
⊢ (𝐹 Fn 𝑋 → Fun 𝐹) | 
| 29 | 22, 27, 28 | 3syl 18 | . . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → Fun 𝐹) | 
| 30 | 13 | adantr 480 | . . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽))) | 
| 31 |  | eqid 2736 | . . . . . . . . . 10
⊢ ∪ (KQ‘𝐽) = ∪
(KQ‘𝐽) | 
| 32 | 31 | cldss 23038 | . . . . . . . . 9
⊢ (𝑤 ∈
(Clsd‘(KQ‘𝐽))
→ 𝑤 ⊆ ∪ (KQ‘𝐽)) | 
| 33 | 30, 32 | syl 17 | . . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑤 ⊆ ∪
(KQ‘𝐽)) | 
| 34 |  | toponuni 22921 | . . . . . . . . 9
⊢
((KQ‘𝐽) ∈
(TopOn‘ran 𝐹) →
ran 𝐹 = ∪ (KQ‘𝐽)) | 
| 35 | 22, 2, 34 | 3syl 18 | . . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ran 𝐹 = ∪
(KQ‘𝐽)) | 
| 36 | 33, 35 | sseqtrrd 4020 | . . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑤 ⊆ ran 𝐹) | 
| 37 |  | funimass1 6647 | . . . . . . 7
⊢ ((Fun
𝐹 ∧ 𝑤 ⊆ ran 𝐹) → ((◡𝐹 “ 𝑤) ⊆ 𝑢 → 𝑤 ⊆ (𝐹 “ 𝑢))) | 
| 38 | 29, 36, 37 | syl2anc 584 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((◡𝐹 “ 𝑤) ⊆ 𝑢 → 𝑤 ⊆ (𝐹 “ 𝑢))) | 
| 39 | 26, 38 | mpd 15 | . . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑤 ⊆ (𝐹 “ 𝑢)) | 
| 40 |  | topontop 22920 | . . . . . . . . . 10
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | 
| 41 | 22, 40 | syl 17 | . . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝐽 ∈ Top) | 
| 42 |  | elssuni 4936 | . . . . . . . . . 10
⊢ (𝑢 ∈ 𝐽 → 𝑢 ⊆ ∪ 𝐽) | 
| 43 | 42 | ad2antrl 728 | . . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑢 ⊆ ∪ 𝐽) | 
| 44 |  | eqid 2736 | . . . . . . . . . 10
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 45 | 44 | clscld 23056 | . . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽)
→ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) | 
| 46 | 41, 43, 45 | syl2anc 584 | . . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) | 
| 47 | 1 | kqcld 23744 | . . . . . . . 8
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽))) | 
| 48 | 22, 46, 47 | syl2anc 584 | . . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽))) | 
| 49 | 44 | sscls 23065 | . . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽)
→ 𝑢 ⊆
((cls‘𝐽)‘𝑢)) | 
| 50 | 41, 43, 49 | syl2anc 584 | . . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢)) | 
| 51 |  | imass2 6119 | . . . . . . . 8
⊢ (𝑢 ⊆ ((cls‘𝐽)‘𝑢) → (𝐹 “ 𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) | 
| 52 | 50, 51 | syl 17 | . . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → (𝐹 “ 𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) | 
| 53 | 31 | clsss2 23081 | . . . . . . 7
⊢ (((𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹 “ 𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) | 
| 54 | 48, 52, 53 | syl2anc 584 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) | 
| 55 |  | simprrr 781 | . . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)) | 
| 56 | 44 | clsss3 23068 | . . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽)
→ ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽) | 
| 57 | 41, 43, 56 | syl2anc 584 | . . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽) | 
| 58 |  | fndm 6670 | . . . . . . . . . . 11
⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋) | 
| 59 | 22, 27, 58 | 3syl 18 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → dom 𝐹 = 𝑋) | 
| 60 |  | toponuni 22921 | . . . . . . . . . . 11
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | 
| 61 | 22, 60 | syl 17 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑋 = ∪ 𝐽) | 
| 62 | 59, 61 | eqtrd 2776 | . . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → dom 𝐹 = ∪ 𝐽) | 
| 63 | 57, 62 | sseqtrrd 4020 | . . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ dom 𝐹) | 
| 64 |  | funimass3 7073 | . . . . . . . 8
⊢ ((Fun
𝐹 ∧ ((cls‘𝐽)‘𝑢) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧 ↔ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧))) | 
| 65 | 29, 63, 64 | syl2anc 584 | . . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧 ↔ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧))) | 
| 66 | 55, 65 | mpbird 257 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧) | 
| 67 | 54, 66 | sstrd 3993 | . . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧) | 
| 68 |  | sseq2 4009 | . . . . . . 7
⊢ (𝑚 = (𝐹 “ 𝑢) → (𝑤 ⊆ 𝑚 ↔ 𝑤 ⊆ (𝐹 “ 𝑢))) | 
| 69 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑚 = (𝐹 “ 𝑢) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢))) | 
| 70 | 69 | sseq1d 4014 | . . . . . . 7
⊢ (𝑚 = (𝐹 “ 𝑢) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧 ↔ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧)) | 
| 71 | 68, 70 | anbi12d 632 | . . . . . 6
⊢ (𝑚 = (𝐹 “ 𝑢) → ((𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧) ↔ (𝑤 ⊆ (𝐹 “ 𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧))) | 
| 72 | 71 | rspcev 3621 | . . . . 5
⊢ (((𝐹 “ 𝑢) ∈ (KQ‘𝐽) ∧ (𝑤 ⊆ (𝐹 “ 𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧)) | 
| 73 | 25, 39, 67, 72 | syl12anc 836 | . . . 4
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧)) | 
| 74 | 21, 73 | rexlimddv 3160 | . . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧)) | 
| 75 | 74 | ralrimivva 3201 | . 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → ∀𝑧 ∈ (KQ‘𝐽)∀𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧)∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧)) | 
| 76 |  | isnrm 23344 | . 2
⊢
((KQ‘𝐽) ∈
Nrm ↔ ((KQ‘𝐽)
∈ Top ∧ ∀𝑧
∈ (KQ‘𝐽)∀𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧)∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))) | 
| 77 | 5, 75, 76 | sylanbrc 583 | 1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm) |