Theorem isopn3 21346

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 21316	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆))
3		inss2 4121	. . . . . . . 8 ⊢ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝒫 𝑆
4	3	unissi 4762	. . . . . . 7 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ ∪ 𝒫 𝑆
5		unipw 5227	. . . . . . 7 ⊢ ∪ 𝒫 𝑆 = 𝑆
6	4, 5	sseqtri 3919	. . . . . 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 4462	. . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ 𝒫 𝑆)
10	8, 9	elind 4087	. . . . . 6 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ (𝐽 ∩ 𝒫 𝑆))
11		elssuni 4768	. . . . . 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 3901	. . . 4 ⊢ (𝑆 ∈ 𝐽 → ∪ (𝐽 ∩ 𝒫 𝑆) = 𝑆)
14	2, 13	sylan9eq 2849	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆)
15	14	ex 413	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 → ((int‘𝐽)‘𝑆) = 𝑆))
16	1	ntropn 21329	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽)
17		eleq1 2868	. . 3 ⊢ (((int‘𝐽)‘𝑆) = 𝑆 → (((int‘𝐽)‘𝑆) ∈ 𝐽 ↔ 𝑆 ∈ 𝐽))
18	16, 17	syl5ibcom 246	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = 𝑆 → 𝑆 ∈ 𝐽))
19	15, 18	impbid 213	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1520   ∈ wcel 2079   ∩ cin 3853   ⊆ wss 3854  𝒫 cpw 4447  ∪ cuni 4739  ‘cfv 6217  Topctop 21173  intcnt 21297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-top 21174  df-ntr 21300

This theorem is referenced by:  ntridm  21348  ntrtop  21350  ntr0  21361  isopn3i  21362  opnnei  21400  cnntr  21555  llycmpkgen2  21830  dvnres  24199  dvcnvre  24287  taylthlem2  24633  ulmdvlem3  24661  abelth  24700  opnbnd  33227  ioontr  41283  cncfuni  41664  fperdvper  41698  dirkercncflem3  41886  dirkercncflem4  41887  fourierdlem58  41945  fourierdlem73  41960