Theorem isopn3 22953

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 22923	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
3		inss2 4201	. . . . . . . 8 ⊢ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝒫 𝑆
4	3	unissi 4880	. . . . . . 7 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ ∪ 𝒫 𝑆
5		unipw 5410	. . . . . . 7 ⊢ ∪ 𝒫 𝑆 = 𝑆
6	4, 5	sseqtri 3995	. . . . . 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 4583	. . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ 𝒫 𝑆)
10	8, 9	elind 4163	. . . . . 6 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ (𝐽 ∩ 𝒫 𝑆))
11		elssuni 4901	. . . . . 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 3964	. . . 4 ⊢ (𝑆 ∈ 𝐽 → ∪ (𝐽 ∩ 𝒫 𝑆) = 𝑆)
14	2, 13	sylan9eq 2784	. . 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 22936	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
17		eleq1 2816	. . 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 2109   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871  ‘cfv 6511  Topctop 22780  intcnt 22904

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-top 22781  df-ntr 22907

This theorem is referenced by:  ntridm  22955  ntrtop  22957  ntr0  22968  isopn3i  22969  opnnei  23007  cnntr  23162  llycmpkgen2  23437  dvnres  25833  dvcnvre  25924  taylthlem2  26282  taylthlem2OLD  26283  ulmdvlem3  26311  abelth  26351  opnbnd  36313  ioontr  45509  cncfuni  45884  fperdvper  45917  dirkercncflem3  46103  dirkercncflem4  46104  fourierdlem58  46162  fourierdlem73  46177