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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 479 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽) | |
2 | elssuni 4702 | . . 3 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ∪ 𝐽) | |
3 | eqid 2778 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | isopn3 21278 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
5 | 2, 4 | sylan2 586 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
6 | 1, 5 | mpbid 224 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 ∪ cuni 4671 ‘cfv 6135 Topctop 21105 intcnt 21229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-top 21106 df-ntr 21232 |
This theorem is referenced by: maxlp 21359 cnntr 21487 bcth2 23536 dvrec 24155 dvmptres 24163 dvcnvlem 24176 dvlip 24193 dvlipcn 24194 dvlip2 24195 dvne0 24211 lhop2 24215 lhop 24216 psercn 24617 dvlog 24834 dvlog2 24836 cxpcn3 24929 efrlim 25148 lgamgulmlem2 25208 cvmlift2lem11 31894 cvmlift2lem12 31895 binomcxplemdvbinom 39512 binomcxplemnotnn0 39515 limciccioolb 40765 limcicciooub 40781 limcresiooub 40786 limcresioolb 40787 dirkercncflem2 41252 fourierdlem32 41287 fourierdlem33 41288 fourierdlem48 41302 fourierdlem49 41303 fourierdlem62 41316 fouriersw 41379 |
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