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Mirrors > Home > MPE Home > Th. List > Mathboxes > restclsseplem | Structured version Visualization version GIF version |
Description: Lemma for restclssep 47034. (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 47032 | . . . 4 ⊢ (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
7 | 6 | ineq1d 4172 | . . 3 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∩ 𝑇)) |
8 | inass 4180 | . . 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 4176 | . . . 4 ⊢ (𝑇 ⊆ 𝑌 ↔ (𝑌 ∩ 𝑇) = 𝑇) | |
13 | 11, 12 | sylib 217 | . . 3 ⊢ (𝜑 → (𝑌 ∩ 𝑇) = 𝑇) |
14 | 13 | ineq2d 4173 | . 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 1542 ∈ wcel 2107 ∩ cin 3910 ⊆ wss 3911 ∅c0 4283 ∪ cuni 4866 ‘cfv 6497 (class class class)co 7358 ↾t crest 17307 Topctop 22258 Clsdccld 22383 clsccl 22385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-en 8887 df-fin 8890 df-fi 9352 df-rest 17309 df-topgen 17330 df-top 22259 df-topon 22276 df-bases 22312 df-cld 22386 df-cls 22388 |
This theorem is referenced by: restclssep 47034 |
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