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Mirrors > Home > MPE Home > Th. List > Mathboxes > restclsseplem | Structured version Visualization version GIF version |
Description: Lemma for restclssep 48633. (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 48631 | . . . 4 ⊢ (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
7 | 6 | ineq1d 4227 | . . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∩ 𝑇)) |
8 | inass 4236 | . . 3 ⊢ ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∩ 𝑇) = (((cls‘𝐽)‘𝑆) ∩ (𝑌 ∩ 𝑇)) | |
9 | 7, 8 | eqtrdi 2789 | . 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 4231 | . . . 4 ⊢ (𝑇 ⊆ 𝑌 ↔ (𝑌 ∩ 𝑇) = 𝑇) | |
13 | 11, 12 | sylib 218 | . . 3 ⊢ (𝜑 → (𝑌 ∩ 𝑇) = 𝑇) |
14 | 13 | ineq2d 4228 | . 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ (𝑌 ∩ 𝑇)) = (((cls‘𝐽)‘𝑆) ∩ 𝑇)) |
15 | 9, 10, 14 | 3eqtr3rd 2782 | 1 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4914 ‘cfv 6558 (class class class)co 7425 ↾t crest 17456 Topctop 22896 Clsdccld 23021 clsccl 23023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-int 4954 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-1st 8007 df-2nd 8008 df-en 8979 df-fin 8982 df-fi 9442 df-rest 17458 df-topgen 17479 df-top 22897 df-topon 22914 df-bases 22950 df-cld 23024 df-cls 23026 |
This theorem is referenced by: restclssep 48633 |
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