Step | Hyp | Ref
| Expression |
1 | | simp3 1130 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑌) |
2 | 1 | ssdifssd 4116 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑆 ∖ {𝑥}) ⊆ 𝑌) |
3 | | restcls.1 |
. . . . . . 7
⊢ 𝑋 = ∪
𝐽 |
4 | | restcls.2 |
. . . . . . 7
⊢ 𝐾 = (𝐽 ↾t 𝑌) |
5 | 3, 4 | restcls 21717 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌)) |
6 | 2, 5 | syld3an3 1401 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) = (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌)) |
7 | 6 | eleq2d 2895 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) ↔ 𝑥 ∈ (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌))) |
8 | | elin 4166 |
. . . 4
⊢ (𝑥 ∈ (((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∩ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥 ∈ 𝑌)) |
9 | 7, 8 | syl6bb 288 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥 ∈ 𝑌))) |
10 | | simp1 1128 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top) |
11 | 3 | toptopon 21453 |
. . . . . . . 8
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
12 | 10, 11 | sylib 219 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ (TopOn‘𝑋)) |
13 | | simp2 1129 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 ⊆ 𝑋) |
14 | | resttopon 21697 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌)) |
15 | 12, 13, 14 | syl2anc 584 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌)) |
16 | 4, 15 | eqeltrid 2914 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ (TopOn‘𝑌)) |
17 | | topontop 21449 |
. . . . 5
⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top) |
18 | 16, 17 | syl 17 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ Top) |
19 | | toponuni 21450 |
. . . . . 6
⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾) |
20 | 16, 19 | syl 17 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 = ∪ 𝐾) |
21 | 1, 20 | sseqtrd 4004 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ∪ 𝐾) |
22 | | eqid 2818 |
. . . . 5
⊢ ∪ 𝐾 =
∪ 𝐾 |
23 | 22 | islp 21676 |
. . . 4
⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐾)
→ (𝑥 ∈
((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})))) |
24 | 18, 21, 23 | syl2anc 584 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐾)‘(𝑆 ∖ {𝑥})))) |
25 | | elin 4166 |
. . . 4
⊢ (𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌) ↔ (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ∧ 𝑥 ∈ 𝑌)) |
26 | 1, 13 | sstrd 3974 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋) |
27 | 3 | islp 21676 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))) |
28 | 10, 26, 27 | syl2anc 584 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))) |
29 | 28 | anbi1d 629 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((𝑥 ∈ ((limPt‘𝐽)‘𝑆) ∧ 𝑥 ∈ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥 ∈ 𝑌))) |
30 | 25, 29 | syl5bb 284 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌) ↔ (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ∧ 𝑥 ∈ 𝑌))) |
31 | 9, 24, 30 | 3bitr4d 312 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝑥 ∈ ((limPt‘𝐾)‘𝑆) ↔ 𝑥 ∈ (((limPt‘𝐽)‘𝑆) ∩ 𝑌))) |
32 | 31 | eqrdv 2816 |
1
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((limPt‘𝐾)‘𝑆) = (((limPt‘𝐽)‘𝑆) ∩ 𝑌)) |