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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 487 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → 𝑆 ∈ 𝐽) | |
2 | elssuni 4870 | . . 3 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ∪ 𝐽) | |
3 | eqid 2823 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | isopn3 21676 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
5 | 2, 4 | sylan2 594 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
6 | 1, 5 | mpbid 234 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∪ cuni 4840 ‘cfv 6357 Topctop 21503 intcnt 21627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-top 21504 df-ntr 21630 |
This theorem is referenced by: maxlp 21757 cnntr 21885 bcth2 23935 dvrec 24554 dvmptres 24562 dvcnvlem 24575 dvlip 24592 dvlipcn 24593 dvlip2 24594 dvne0 24610 lhop2 24614 lhop 24615 psercn 25016 dvlog 25236 dvlog2 25238 cxpcn3 25331 efrlim 25549 lgamgulmlem2 25609 cvmlift2lem11 32562 cvmlift2lem12 32563 binomcxplemdvbinom 40692 binomcxplemnotnn0 40695 limciccioolb 41909 limcicciooub 41925 limcresiooub 41930 limcresioolb 41931 dirkercncflem2 42396 fourierdlem32 42431 fourierdlem33 42432 fourierdlem48 42446 fourierdlem49 42447 fourierdlem62 42460 fouriersw 42523 |
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