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Mirrors > Home > ILE Home > Th. List > ntrss2 | 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 12279 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆)) |
3 | inss2 3297 | . . . 4 ⊢ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝒫 𝑆 | |
4 | 3 | unissi 3759 | . . 3 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ ∪ 𝒫 𝑆 |
5 | unipw 4139 | . . 3 ⊢ ∪ 𝒫 𝑆 = 𝑆 | |
6 | 4, 5 | sseqtri 3131 | . 2 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝑆 |
7 | 2, 6 | eqsstrdi 3149 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∩ cin 3070 ⊆ wss 3071 𝒫 cpw 3510 ∪ cuni 3736 ‘cfv 5123 Topctop 12164 intcnt 12262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-top 12165 df-ntr 12265 |
This theorem is referenced by: ntrin 12293 neiint 12314 topssnei 12331 iscnp4 12387 cnntri 12393 cnntr 12394 cnptoprest 12408 dvbss 12823 dvfgg 12826 dvcnp2cntop 12832 dvaddxxbr 12834 dvmulxxbr 12835 |
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