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Mirrors > Home > MPE Home > Th. List > isopn3i | Structured version Visualization version GIF version |
Description: An open subset equals its own interior. (Contributed by Mario Carneiro, 30-Dec-2016.) |
Ref | Expression |
---|---|
isopn3i | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽) | |
2 | elssuni 4830 | . . 3 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ∪ 𝐽) | |
3 | eqid 2798 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | isopn3 21671 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
5 | 2, 4 | sylan2 595 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
6 | 1, 5 | mpbid 235 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∪ 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: maxlp 21752 cnntr 21880 bcth2 23934 dvrec 24558 dvmptres 24566 dvcnvlem 24579 dvlip 24596 dvlipcn 24597 dvlip2 24598 dvne0 24614 lhop2 24618 lhop 24619 psercn 25021 dvlog 25242 dvlog2 25244 cxpcn3 25337 efrlim 25555 lgamgulmlem2 25615 cvmlift2lem11 32673 cvmlift2lem12 32674 binomcxplemdvbinom 41057 binomcxplemnotnn0 41060 limciccioolb 42263 limcicciooub 42279 limcresiooub 42284 limcresioolb 42285 dirkercncflem2 42746 fourierdlem32 42781 fourierdlem33 42782 fourierdlem48 42796 fourierdlem49 42797 fourierdlem62 42810 fouriersw 42873 |
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