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| Mirrors > Home > MPE Home > Th. List > Mathboxes > restclsseplem | Structured version Visualization version GIF version | ||
| Description: Lemma for restclssep 48786. (Contributed by Zhi Wang, 2-Sep-2024.) |
| Ref | Expression |
|---|---|
| restcls2.1 | ⊢ (𝜑 → 𝐽 ∈ Top) |
| restcls2.2 | ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
| restcls2.3 | ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
| restcls2.4 | ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌)) |
| restcls2.5 | ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾)) |
| restclsseplem.6 | ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) |
| restclsseplem.7 | ⊢ (𝜑 → 𝑇 ⊆ 𝑌) |
| Ref | Expression |
|---|---|
| restclsseplem | ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | restcls2.1 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top) | |
| 2 | restcls2.2 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) | |
| 3 | restcls2.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
| 4 | restcls2.4 | . . . . 5 ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌)) | |
| 5 | restcls2.5 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾)) | |
| 6 | 1, 2, 3, 4, 5 | restcls2 48784 | . . . 4 ⊢ (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
| 7 | 6 | ineq1d 4218 | . . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∩ 𝑇)) |
| 8 | inass 4227 | . . 3 ⊢ ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∩ 𝑇) = (((cls‘𝐽)‘𝑆) ∩ (𝑌 ∩ 𝑇)) | |
| 9 | 7, 8 | eqtrdi 2792 | . 2 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = (((cls‘𝐽)‘𝑆) ∩ (𝑌 ∩ 𝑇))) |
| 10 | restclsseplem.6 | . 2 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) | |
| 11 | restclsseplem.7 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑌) | |
| 12 | sseqin2 4222 | . . . 4 ⊢ (𝑇 ⊆ 𝑌 ↔ (𝑌 ∩ 𝑇) = 𝑇) | |
| 13 | 11, 12 | sylib 218 | . . 3 ⊢ (𝜑 → (𝑌 ∩ 𝑇) = 𝑇) |
| 14 | 13 | ineq2d 4219 | . 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ (𝑌 ∩ 𝑇)) = (((cls‘𝐽)‘𝑆) ∩ 𝑇)) |
| 15 | 9, 10, 14 | 3eqtr3rd 2785 | 1 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 ∪ cuni 4905 ‘cfv 6559 (class class class)co 7429 ↾t crest 17461 Topctop 22889 Clsdccld 23014 clsccl 23016 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-en 8982 df-fin 8985 df-fi 9447 df-rest 17463 df-topgen 17484 df-top 22890 df-topon 22907 df-bases 22943 df-cld 23017 df-cls 23019 |
| This theorem is referenced by: restclssep 48786 |
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