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Mirrors > Home > MPE Home > Th. List > ntrss2 | Structured version Visualization version GIF version |
Description: A subset includes its interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
ntrss2 | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | ntrval 21641 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆)) |
3 | inss2 4156 | . . . 4 ⊢ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝒫 𝑆 | |
4 | 3 | unissi 4809 | . . 3 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ ∪ 𝒫 𝑆 |
5 | unipw 5308 | . . 3 ⊢ ∪ 𝒫 𝑆 = 𝑆 | |
6 | 4, 5 | sseqtri 3951 | . 2 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝑆 |
7 | 2, 6 | eqsstrdi 3969 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 ‘cfv 6324 Topctop 21498 intcnt 21622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-top 21499 df-ntr 21625 |
This theorem is referenced by: ntrin 21666 neiint 21709 opnnei 21725 topssnei 21729 maxlp 21752 restntr 21787 iscnp4 21868 cnntri 21876 cnntr 21880 cnprest 21894 llycmpkgen2 22155 xkococnlem 22264 flimopn 22580 fclsneii 22622 fcfnei 22640 subgntr 22712 iccntr 23426 rectbntr0 23437 bcthlem5 23932 limcflf 24484 dvbss 24504 perfdvf 24506 dvreslem 24512 dvcnp2 24523 dvnres 24534 dvaddbr 24541 dvcmulf 24548 dvmptres2 24565 dvmptcmul 24567 dvmptntr 24574 dvcnvre 24622 taylthlem1 24968 taylthlem2 24969 ulmdvlem3 24997 lgamucov2 25624 ubthlem1 28653 kur14lem6 32571 cvmlift2lem12 32674 opnbnd 33786 opnregcld 33791 cldregopn 33792 dvresntr 42560 |
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