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Theorem isopn3 21674
Description: A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
clscld.1 𝑋 = 𝐽
Assertion
Ref Expression
isopn3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))

Proof of Theorem isopn3
StepHypRef Expression
1 clscld.1 . . . . 5 𝑋 = 𝐽
21ntrval 21644 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
3 inss2 4191 . . . . . . . 8 (𝐽 ∩ 𝒫 𝑆) ⊆ 𝒫 𝑆
43unissi 4833 . . . . . . 7 (𝐽 ∩ 𝒫 𝑆) ⊆ 𝒫 𝑆
5 unipw 5330 . . . . . . 7 𝒫 𝑆 = 𝑆
64, 5sseqtri 3989 . . . . . 6 (𝐽 ∩ 𝒫 𝑆) ⊆ 𝑆
76a1i 11 . . . . 5 (𝑆𝐽 (𝐽 ∩ 𝒫 𝑆) ⊆ 𝑆)
8 id 22 . . . . . . 7 (𝑆𝐽𝑆𝐽)
9 pwidg 4544 . . . . . . 7 (𝑆𝐽𝑆 ∈ 𝒫 𝑆)
108, 9elind 4156 . . . . . 6 (𝑆𝐽𝑆 ∈ (𝐽 ∩ 𝒫 𝑆))
11 elssuni 4854 . . . . . 6 (𝑆 ∈ (𝐽 ∩ 𝒫 𝑆) → 𝑆 (𝐽 ∩ 𝒫 𝑆))
1210, 11syl 17 . . . . 5 (𝑆𝐽𝑆 (𝐽 ∩ 𝒫 𝑆))
137, 12eqssd 3970 . . . 4 (𝑆𝐽 (𝐽 ∩ 𝒫 𝑆) = 𝑆)
142, 13sylan9eq 2879 . . 3 (((𝐽 ∈ Top ∧ 𝑆𝑋) ∧ 𝑆𝐽) → ((int‘𝐽)‘𝑆) = 𝑆)
1514ex 416 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 → ((int‘𝐽)‘𝑆) = 𝑆))
161ntropn 21657 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
17 eleq1 2903 . . 3 (((int‘𝐽)‘𝑆) = 𝑆 → (((int‘𝐽)‘𝑆) ∈ 𝐽𝑆𝐽))
1816, 17syl5ibcom 248 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (((int‘𝐽)‘𝑆) = 𝑆𝑆𝐽))
1915, 18impbid 215 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cin 3918  wss 3919  𝒫 cpw 4522   cuni 4824  cfv 6343  Topctop 21501  intcnt 21625
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-top 21502  df-ntr 21628
This theorem is referenced by:  ntridm  21676  ntrtop  21678  ntr0  21689  isopn3i  21690  opnnei  21728  cnntr  21883  llycmpkgen2  22158  dvnres  24537  dvcnvre  24625  taylthlem2  24972  ulmdvlem3  25000  abelth  25039  opnbnd  33730  ioontr  42074  cncfuni  42454  fperdvper  42487  dirkercncflem3  42673  dirkercncflem4  42674  fourierdlem58  42732  fourierdlem73  42747
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