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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 484 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽) | |
| 2 | elssuni 4937 | . . 3 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ∪ 𝐽) | |
| 3 | eqid 2737 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 4 | 3 | isopn3 23074 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
| 5 | 2, 4 | sylan2 593 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
| 6 | 1, 5 | mpbid 232 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 intcnt 23025 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-top 22900 df-ntr 23028 |
| This theorem is referenced by: maxlp 23155 cnntr 23283 bcth2 25364 dvrec 25993 dvmptres 26001 dvcnvlem 26014 dvlip 26032 dvlipcn 26033 dvlip2 26034 dvne0 26050 lhop2 26054 lhop 26055 psercn 26470 dvlog 26693 dvlog2 26695 cxpcn3 26791 efrlim 27012 efrlimOLD 27013 lgamgulmlem2 27073 cvmlift2lem11 35318 cvmlift2lem12 35319 dvrelog3 42066 redvmptabs 42390 binomcxplemdvbinom 44372 binomcxplemnotnn0 44375 limciccioolb 45636 limcicciooub 45652 limcresiooub 45657 limcresioolb 45658 dirkercncflem2 46119 fourierdlem32 46154 fourierdlem33 46155 fourierdlem48 46169 fourierdlem49 46170 fourierdlem62 46183 fouriersw 46246 |
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