| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | topontop 22919 | . . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | 
| 2 | 1 | adantr 480 | . 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Top) | 
| 3 |  | simplr 769 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (KQ‘𝐽) ∈ Reg) | 
| 4 |  | simpll 767 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 5 |  | simprl 771 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑧 ∈ 𝐽) | 
| 6 |  | kqval.2 | . . . . . . 7
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | 
| 7 | 6 | kqopn 23742 | . . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽)) | 
| 8 | 4, 5, 7 | syl2anc 584 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (𝐹 “ 𝑧) ∈ (KQ‘𝐽)) | 
| 9 |  | simprr 773 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑤 ∈ 𝑧) | 
| 10 |  | toponss 22933 | . . . . . . . . 9
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋) | 
| 11 | 4, 5, 10 | syl2anc 584 | . . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑧 ⊆ 𝑋) | 
| 12 | 11, 9 | sseldd 3984 | . . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → 𝑤 ∈ 𝑋) | 
| 13 | 6 | kqfvima 23738 | . . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝑧 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑧))) | 
| 14 | 4, 5, 12, 13 | syl3anc 1373 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (𝑤 ∈ 𝑧 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑧))) | 
| 15 | 9, 14 | mpbid 232 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → (𝐹‘𝑤) ∈ (𝐹 “ 𝑧)) | 
| 16 |  | regsep 23342 | . . . . 5
⊢
(((KQ‘𝐽)
∈ Reg ∧ (𝐹 “
𝑧) ∈ (KQ‘𝐽) ∧ (𝐹‘𝑤) ∈ (𝐹 “ 𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧))) | 
| 17 | 3, 8, 15, 16 | syl3anc 1373 | . . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → ∃𝑛 ∈ (KQ‘𝐽)((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧))) | 
| 18 | 4 | adantr 480 | . . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 19 | 6 | kqid 23736 | . . . . . . 7
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽))) | 
| 20 | 18, 19 | syl 17 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽))) | 
| 21 |  | simprl 771 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑛 ∈ (KQ‘𝐽)) | 
| 22 |  | cnima 23273 | . . . . . 6
⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑛 ∈ (KQ‘𝐽)) → (◡𝐹 “ 𝑛) ∈ 𝐽) | 
| 23 | 20, 21, 22 | syl2anc 584 | . . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (◡𝐹 “ 𝑛) ∈ 𝐽) | 
| 24 | 12 | adantr 480 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ∈ 𝑋) | 
| 25 |  | simprrl 781 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (𝐹‘𝑤) ∈ 𝑛) | 
| 26 | 6 | kqffn 23733 | . . . . . . 7
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋) | 
| 27 |  | elpreima 7078 | . . . . . . 7
⊢ (𝐹 Fn 𝑋 → (𝑤 ∈ (◡𝐹 “ 𝑛) ↔ (𝑤 ∈ 𝑋 ∧ (𝐹‘𝑤) ∈ 𝑛))) | 
| 28 | 18, 26, 27 | 3syl 18 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (𝑤 ∈ (◡𝐹 “ 𝑛) ↔ (𝑤 ∈ 𝑋 ∧ (𝐹‘𝑤) ∈ 𝑛))) | 
| 29 | 24, 25, 28 | mpbir2and 713 | . . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑤 ∈ (◡𝐹 “ 𝑛)) | 
| 30 | 6 | kqtopon 23735 | . . . . . . . . . 10
⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹)) | 
| 31 |  | topontop 22919 | . . . . . . . . . 10
⊢
((KQ‘𝐽) ∈
(TopOn‘ran 𝐹) →
(KQ‘𝐽) ∈
Top) | 
| 32 | 18, 30, 31 | 3syl 18 | . . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (KQ‘𝐽) ∈ Top) | 
| 33 |  | elssuni 4937 | . . . . . . . . . 10
⊢ (𝑛 ∈ (KQ‘𝐽) → 𝑛 ⊆ ∪
(KQ‘𝐽)) | 
| 34 | 33 | ad2antrl 728 | . . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑛 ⊆ ∪
(KQ‘𝐽)) | 
| 35 |  | eqid 2737 | . . . . . . . . . 10
⊢ ∪ (KQ‘𝐽) = ∪
(KQ‘𝐽) | 
| 36 | 35 | clscld 23055 | . . . . . . . . 9
⊢
(((KQ‘𝐽)
∈ Top ∧ 𝑛 ⊆
∪ (KQ‘𝐽)) → ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽))) | 
| 37 | 32, 34, 36 | syl2anc 584 | . . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽))) | 
| 38 |  | cnclima 23276 | . . . . . . . 8
⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ ((cls‘(KQ‘𝐽))‘𝑛) ∈ (Clsd‘(KQ‘𝐽))) → (◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽)) | 
| 39 | 20, 37, 38 | syl2anc 584 | . . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽)) | 
| 40 | 35 | sscls 23064 | . . . . . . . . 9
⊢
(((KQ‘𝐽)
∈ Top ∧ 𝑛 ⊆
∪ (KQ‘𝐽)) → 𝑛 ⊆ ((cls‘(KQ‘𝐽))‘𝑛)) | 
| 41 | 32, 34, 40 | syl2anc 584 | . . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑛 ⊆ ((cls‘(KQ‘𝐽))‘𝑛)) | 
| 42 |  | imass2 6120 | . . . . . . . 8
⊢ (𝑛 ⊆
((cls‘(KQ‘𝐽))‘𝑛) → (◡𝐹 “ 𝑛) ⊆ (◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛))) | 
| 43 | 41, 42 | syl 17 | . . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (◡𝐹 “ 𝑛) ⊆ (◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛))) | 
| 44 |  | eqid 2737 | . . . . . . . 8
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 45 | 44 | clsss2 23080 | . . . . . . 7
⊢ (((◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ∈ (Clsd‘𝐽) ∧ (◡𝐹 “ 𝑛) ⊆ (◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛))) → ((cls‘𝐽)‘(◡𝐹 “ 𝑛)) ⊆ (◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛))) | 
| 46 | 39, 43, 45 | syl2anc 584 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → ((cls‘𝐽)‘(◡𝐹 “ 𝑛)) ⊆ (◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛))) | 
| 47 |  | simprrr 782 | . . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)) | 
| 48 |  | imass2 6120 | . . . . . . . 8
⊢
(((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧) → (◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ⊆ (◡𝐹 “ (𝐹 “ 𝑧))) | 
| 49 | 47, 48 | syl 17 | . . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ⊆ (◡𝐹 “ (𝐹 “ 𝑧))) | 
| 50 | 5 | adantr 480 | . . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → 𝑧 ∈ 𝐽) | 
| 51 | 6 | kqsat 23739 | . . . . . . . 8
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → (◡𝐹 “ (𝐹 “ 𝑧)) = 𝑧) | 
| 52 | 18, 50, 51 | syl2anc 584 | . . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (◡𝐹 “ (𝐹 “ 𝑧)) = 𝑧) | 
| 53 | 49, 52 | sseqtrd 4020 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → (◡𝐹 “ ((cls‘(KQ‘𝐽))‘𝑛)) ⊆ 𝑧) | 
| 54 | 46, 53 | sstrd 3994 | . . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → ((cls‘𝐽)‘(◡𝐹 “ 𝑛)) ⊆ 𝑧) | 
| 55 |  | eleq2 2830 | . . . . . . 7
⊢ (𝑚 = (◡𝐹 “ 𝑛) → (𝑤 ∈ 𝑚 ↔ 𝑤 ∈ (◡𝐹 “ 𝑛))) | 
| 56 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑚 = (◡𝐹 “ 𝑛) → ((cls‘𝐽)‘𝑚) = ((cls‘𝐽)‘(◡𝐹 “ 𝑛))) | 
| 57 | 56 | sseq1d 4015 | . . . . . . 7
⊢ (𝑚 = (◡𝐹 “ 𝑛) → (((cls‘𝐽)‘𝑚) ⊆ 𝑧 ↔ ((cls‘𝐽)‘(◡𝐹 “ 𝑛)) ⊆ 𝑧)) | 
| 58 | 55, 57 | anbi12d 632 | . . . . . 6
⊢ (𝑚 = (◡𝐹 “ 𝑛) → ((𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧) ↔ (𝑤 ∈ (◡𝐹 “ 𝑛) ∧ ((cls‘𝐽)‘(◡𝐹 “ 𝑛)) ⊆ 𝑧))) | 
| 59 | 58 | rspcev 3622 | . . . . 5
⊢ (((◡𝐹 “ 𝑛) ∈ 𝐽 ∧ (𝑤 ∈ (◡𝐹 “ 𝑛) ∧ ((cls‘𝐽)‘(◡𝐹 “ 𝑛)) ⊆ 𝑧)) → ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧)) | 
| 60 | 23, 29, 54, 59 | syl12anc 837 | . . . 4
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) ∧ (𝑛 ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑤) ∈ 𝑛 ∧ ((cls‘(KQ‘𝐽))‘𝑛) ⊆ (𝐹 “ 𝑧)))) → ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧)) | 
| 61 | 17, 60 | rexlimddv 3161 | . . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) ∧ (𝑧 ∈ 𝐽 ∧ 𝑤 ∈ 𝑧)) → ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧)) | 
| 62 | 61 | ralrimivva 3202 | . 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) →
∀𝑧 ∈ 𝐽 ∀𝑤 ∈ 𝑧 ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧)) | 
| 63 |  | isreg 23340 | . 2
⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑧 ∈ 𝐽 ∀𝑤 ∈ 𝑧 ∃𝑚 ∈ 𝐽 (𝑤 ∈ 𝑚 ∧ ((cls‘𝐽)‘𝑚) ⊆ 𝑧))) | 
| 64 | 2, 62, 63 | sylanbrc 583 | 1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Reg) → 𝐽 ∈ Reg) |