Theorem isopn3 23004

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 ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))

Proof of Theorem isopn3

Step	Hyp	Ref	Expression
1		clscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2	1	ntrval 22974	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
3		inss2 4213	. . . . . . . 8 ⊢ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝒫 𝑆
4	3	unissi 4892	. . . . . . 7 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ ∪ 𝒫 𝑆
5		unipw 5425	. . . . . . 7 ⊢ ∪ 𝒫 𝑆 = 𝑆
6	4, 5	sseqtri 4007	. . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝑆
7	6	a1i 11	. . . . 5 ⊢ (𝑆 ∈ 𝐽 → ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝑆)
8		id 22	. . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ 𝐽)
9		pwidg 4595	. . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ 𝒫 𝑆)
10	8, 9	elind 4175	. . . . . 6 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ (𝐽 ∩ 𝒫 𝑆))
11		elssuni 4913	. . . . . 6 ⊢ (𝑆 ∈ (𝐽 ∩ 𝒫 𝑆) → 𝑆 ⊆ ∪ (𝐽 ∩ 𝒫 𝑆))
12	10, 11	syl 17	. . . . 5 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ∪ (𝐽 ∩ 𝒫 𝑆))
13	7, 12	eqssd 3976	. . . 4 ⊢ (𝑆 ∈ 𝐽 → ∪ (𝐽 ∩ 𝒫 𝑆) = 𝑆)
14	2, 13	sylan9eq 2790	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆)
15	14	ex 412	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 → ((int‘𝐽)‘𝑆) = 𝑆))
16	1	ntropn 22987	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
17		eleq1 2822	. . 3 ⊢ (((int‘𝐽)‘𝑆) = 𝑆 → (((int‘𝐽)‘𝑆) ∈ 𝐽 ↔ 𝑆 ∈ 𝐽))
18	16, 17	syl5ibcom 245	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = 𝑆 → 𝑆 ∈ 𝐽))
19	15, 18	impbid 212	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∩ cin 3925   ⊆ wss 3926  𝒫 cpw 4575  ∪ cuni 4883  ‘cfv 6531  Topctop 22831  intcnt 22955

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-top 22832  df-ntr 22958

This theorem is referenced by:  ntridm  23006  ntrtop  23008  ntr0  23019  isopn3i  23020  opnnei  23058  cnntr  23213  llycmpkgen2  23488  dvnres  25885  dvcnvre  25976  taylthlem2  26334  taylthlem2OLD  26335  ulmdvlem3  26363  abelth  26403  opnbnd  36343  ioontr  45540  cncfuni  45915  fperdvper  45948  dirkercncflem3  46134  dirkercncflem4  46135  fourierdlem58  46193  fourierdlem73  46208