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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 483 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽) | |
2 | elssuni 4942 | . . 3 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ∪ 𝐽) | |
3 | eqid 2727 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | isopn3 22988 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
5 | 2, 4 | sylan2 591 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
6 | 1, 5 | mpbid 231 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3947 ∪ cuni 4910 ‘cfv 6551 Topctop 22813 intcnt 22939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-top 22814 df-ntr 22942 |
This theorem is referenced by: maxlp 23069 cnntr 23197 bcth2 25276 dvrec 25905 dvmptres 25913 dvcnvlem 25926 dvlip 25944 dvlipcn 25945 dvlip2 25946 dvne0 25962 lhop2 25966 lhop 25967 psercn 26381 dvlog 26603 dvlog2 26605 cxpcn3 26701 efrlim 26919 efrlimOLD 26920 lgamgulmlem2 26980 cvmlift2lem11 34928 cvmlift2lem12 34929 dvrelog3 41540 binomcxplemdvbinom 43793 binomcxplemnotnn0 43796 limciccioolb 45011 limcicciooub 45027 limcresiooub 45032 limcresioolb 45033 dirkercncflem2 45494 fourierdlem32 45529 fourierdlem33 45530 fourierdlem48 45544 fourierdlem49 45545 fourierdlem62 45558 fouriersw 45621 |
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