| Step | Hyp | Ref
| Expression |
| 1 | | simp1 1137 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐽 ∈ Top) |
| 2 | | sstr 3992 |
. . . . . . . 8
⊢ ((𝑆 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋) → 𝑆 ⊆ 𝑋) |
| 3 | 2 | ancoms 458 |
. . . . . . 7
⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋) |
| 4 | 3 | 3adant1 1131 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑋) |
| 5 | | restcls.1 |
. . . . . . 7
⊢ 𝑋 = ∪
𝐽 |
| 6 | 5 | clscld 23055 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽)) |
| 7 | 1, 4, 6 | syl2anc 584 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽)) |
| 8 | | eqid 2737 |
. . . . 5
⊢
(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌) |
| 9 | | ineq1 4213 |
. . . . . 6
⊢ (𝑥 = ((cls‘𝐽)‘𝑆) → (𝑥 ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
| 10 | 9 | rspceeqv 3645 |
. . . . 5
⊢
((((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥 ∩ 𝑌)) |
| 11 | 7, 8, 10 | sylancl 586 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥 ∩ 𝑌)) |
| 12 | | restcls.2 |
. . . . . . 7
⊢ 𝐾 = (𝐽 ↾t 𝑌) |
| 13 | 12 | fveq2i 6909 |
. . . . . 6
⊢
(Clsd‘𝐾) =
(Clsd‘(𝐽
↾t 𝑌)) |
| 14 | 13 | eleq2i 2833 |
. . . . 5
⊢
((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽 ↾t 𝑌))) |
| 15 | 5 | restcld 23180 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽 ↾t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥 ∩ 𝑌))) |
| 16 | 15 | 3adant3 1133 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽 ↾t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥 ∩ 𝑌))) |
| 17 | 14, 16 | bitrid 283 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥 ∩ 𝑌))) |
| 18 | 11, 17 | mpbird 257 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾)) |
| 19 | 5 | sscls 23064 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) |
| 20 | 1, 4, 19 | syl2anc 584 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) |
| 21 | | simp3 1139 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ 𝑌) |
| 22 | 20, 21 | ssind 4241 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
| 23 | | eqid 2737 |
. . . 4
⊢ ∪ 𝐾 =
∪ 𝐾 |
| 24 | 23 | clsss2 23080 |
. . 3
⊢
(((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ∧ 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
| 25 | 18, 22, 24 | syl2anc 584 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
| 26 | 12 | fveq2i 6909 |
. . . . . 6
⊢
(cls‘𝐾) =
(cls‘(𝐽
↾t 𝑌)) |
| 27 | 26 | fveq1i 6907 |
. . . . 5
⊢
((cls‘𝐾)‘𝑆) = ((cls‘(𝐽 ↾t 𝑌))‘𝑆) |
| 28 | | id 22 |
. . . . . . . . 9
⊢ (𝑌 ⊆ 𝑋 → 𝑌 ⊆ 𝑋) |
| 29 | 5 | topopn 22912 |
. . . . . . . . 9
⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
| 30 | | ssexg 5323 |
. . . . . . . . 9
⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑌 ∈ V) |
| 31 | 28, 29, 30 | syl2anr 597 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V) |
| 32 | | resttop 23168 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽 ↾t 𝑌) ∈ Top) |
| 33 | 31, 32 | syldan 591 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ Top) |
| 34 | 33 | 3adant3 1133 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (𝐽 ↾t 𝑌) ∈ Top) |
| 35 | 5 | restuni 23170 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ (𝐽 ↾t 𝑌)) |
| 36 | 35 | 3adant3 1133 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑌 = ∪ (𝐽 ↾t 𝑌)) |
| 37 | 21, 36 | sseqtrd 4020 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ∪ (𝐽 ↾t 𝑌)) |
| 38 | | eqid 2737 |
. . . . . . 7
⊢ ∪ (𝐽
↾t 𝑌) =
∪ (𝐽 ↾t 𝑌) |
| 39 | 38 | clscld 23055 |
. . . . . 6
⊢ (((𝐽 ↾t 𝑌) ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽
↾t 𝑌))
→ ((cls‘(𝐽
↾t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌))) |
| 40 | 34, 37, 39 | syl2anc 584 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌))) |
| 41 | 27, 40 | eqeltrid 2845 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌))) |
| 42 | 5 | restcld 23180 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) |
| 43 | 42 | 3adant3 1133 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽 ↾t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) |
| 44 | 41, 43 | mpbid 232 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌)) |
| 45 | 12, 33 | eqeltrid 2845 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top) |
| 46 | 45 | 3adant3 1133 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝐾 ∈ Top) |
| 47 | 12 | unieqi 4919 |
. . . . . . . . 9
⊢ ∪ 𝐾 =
∪ (𝐽 ↾t 𝑌) |
| 48 | 47 | eqcomi 2746 |
. . . . . . . 8
⊢ ∪ (𝐽
↾t 𝑌) =
∪ 𝐾 |
| 49 | 48 | sscls 23064 |
. . . . . . 7
⊢ ((𝐾 ∈ Top ∧ 𝑆 ⊆ ∪ (𝐽
↾t 𝑌))
→ 𝑆 ⊆
((cls‘𝐾)‘𝑆)) |
| 50 | 46, 37, 49 | syl2anc 584 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆)) |
| 51 | 50 | adantr 480 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆)) |
| 52 | | inss1 4237 |
. . . . . . 7
⊢ (𝑥 ∩ 𝑌) ⊆ 𝑥 |
| 53 | | sseq1 4009 |
. . . . . . 7
⊢
(((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → (((cls‘𝐾)‘𝑆) ⊆ 𝑥 ↔ (𝑥 ∩ 𝑌) ⊆ 𝑥)) |
| 54 | 52, 53 | mpbiri 258 |
. . . . . 6
⊢
(((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥) |
| 55 | 54 | ad2antll 729 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥) |
| 56 | 51, 55 | sstrd 3994 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → 𝑆 ⊆ 𝑥) |
| 57 | 5 | clsss2 23080 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥) |
| 58 | 57 | adantl 481 |
. . . . . . . . 9
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥) |
| 59 | 58 | ssrind 4244 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥)) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥 ∩ 𝑌)) |
| 60 | | sseq2 4010 |
. . . . . . . 8
⊢
(((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥 ∩ 𝑌))) |
| 61 | 59, 60 | syl5ibrcom 247 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑥)) → (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))) |
| 62 | 61 | expr 456 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆 ⊆ 𝑥 → (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)))) |
| 63 | 62 | com23 86 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌) → (𝑆 ⊆ 𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)))) |
| 64 | 63 | impr 454 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → (𝑆 ⊆ 𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))) |
| 65 | 56, 64 | mpd 15 |
. . 3
⊢ (((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥 ∩ 𝑌))) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)) |
| 66 | 44, 65 | rexlimddv 3161 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)) |
| 67 | 25, 66 | eqssd 4001 |
1
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |