| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | llytop 23480 | . . 3
⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top) | 
| 2 |  | resttop 23168 | . . 3
⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top) | 
| 3 | 1, 2 | sylan 580 | . 2
⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Top) | 
| 4 |  | restopn2 23185 | . . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵))) | 
| 5 | 1, 4 | sylan 580 | . . . 4
⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) ↔ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵))) | 
| 6 |  | simp1l 1198 | . . . . . . . . 9
⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝐽 ∈ Locally 𝐴) | 
| 7 |  | simp2l 1200 | . . . . . . . . 9
⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ 𝐽) | 
| 8 |  | simp3 1139 | . . . . . . . . 9
⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥) | 
| 9 |  | llyi 23482 | . . . . . . . . 9
⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ 𝐽 (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)) | 
| 10 | 6, 7, 8, 9 | syl3anc 1373 | . . . . . . . 8
⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ 𝐽 (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)) | 
| 11 |  | simprl 771 | . . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝐽) | 
| 12 |  | simprr1 1222 | . . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ 𝑥) | 
| 13 |  | simpl2r 1228 | . . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑥 ⊆ 𝐵) | 
| 14 | 12, 13 | sstrd 3994 | . . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ⊆ 𝐵) | 
| 15 | 6, 1 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → 𝐽 ∈ Top) | 
| 16 | 15 | adantr 480 | . . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝐽 ∈ Top) | 
| 17 |  | simpl1r 1226 | . . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝐵 ∈ 𝐽) | 
| 18 |  | restopn2 23185 | . . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ 𝐽) → (𝑣 ∈ (𝐽 ↾t 𝐵) ↔ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝐵))) | 
| 19 | 16, 17, 18 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → (𝑣 ∈ (𝐽 ↾t 𝐵) ↔ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝐵))) | 
| 20 | 11, 14, 19 | mpbir2and 713 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ (𝐽 ↾t 𝐵)) | 
| 21 |  | velpw 4605 | . . . . . . . . . . . . 13
⊢ (𝑣 ∈ 𝒫 𝑥 ↔ 𝑣 ⊆ 𝑥) | 
| 22 | 12, 21 | sylibr 234 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ 𝒫 𝑥) | 
| 23 | 20, 22 | elind 4200 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)) | 
| 24 |  | simprr2 1223 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → 𝑦 ∈ 𝑣) | 
| 25 |  | restabs 23173 | . . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ 𝑣 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐽) → ((𝐽 ↾t 𝐵) ↾t 𝑣) = (𝐽 ↾t 𝑣)) | 
| 26 | 16, 14, 17, 25 | syl3anc 1373 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → ((𝐽 ↾t 𝐵) ↾t 𝑣) = (𝐽 ↾t 𝑣)) | 
| 27 |  | simprr3 1224 | . . . . . . . . . . . 12
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → (𝐽 ↾t 𝑣) ∈ 𝐴) | 
| 28 | 26, 27 | eqeltrd 2841 | . . . . . . . . . . 11
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴) | 
| 29 | 23, 24, 28 | jca32 515 | . . . . . . . . . 10
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) ∧ (𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴))) → (𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴))) | 
| 30 | 29 | ex 412 | . . . . . . . . 9
⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → ((𝑣 ∈ 𝐽 ∧ (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴)) → (𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥) ∧ (𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴)))) | 
| 31 | 30 | reximdv2 3164 | . . . . . . . 8
⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → (∃𝑣 ∈ 𝐽 (𝑣 ⊆ 𝑥 ∧ 𝑦 ∈ 𝑣 ∧ (𝐽 ↾t 𝑣) ∈ 𝐴) → ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴))) | 
| 32 | 10, 31 | mpd 15 | . . . . . . 7
⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴)) | 
| 33 | 32 | 3expa 1119 | . . . . . 6
⊢ ((((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)) ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴)) | 
| 34 | 33 | ralrimiva 3146 | . . . . 5
⊢ (((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) ∧ (𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵)) → ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴)) | 
| 35 | 34 | ex 412 | . . . 4
⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → ((𝑥 ∈ 𝐽 ∧ 𝑥 ⊆ 𝐵) → ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴))) | 
| 36 | 5, 35 | sylbid 240 | . . 3
⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝑥 ∈ (𝐽 ↾t 𝐵) → ∀𝑦 ∈ 𝑥 ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴))) | 
| 37 | 36 | ralrimiv 3145 | . 2
⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → ∀𝑥 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑥 ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴)) | 
| 38 |  | islly 23476 | . 2
⊢ ((𝐽 ↾t 𝐵) ∈ Locally 𝐴 ↔ ((𝐽 ↾t 𝐵) ∈ Top ∧ ∀𝑥 ∈ (𝐽 ↾t 𝐵)∀𝑦 ∈ 𝑥 ∃𝑣 ∈ ((𝐽 ↾t 𝐵) ∩ 𝒫 𝑥)(𝑦 ∈ 𝑣 ∧ ((𝐽 ↾t 𝐵) ↾t 𝑣) ∈ 𝐴))) | 
| 39 | 3, 37, 38 | sylanbrc 583 | 1
⊢ ((𝐽 ∈ Locally 𝐴 ∧ 𝐵 ∈ 𝐽) → (𝐽 ↾t 𝐵) ∈ Locally 𝐴) |