MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ntrval Structured version   Visualization version   GIF version

Theorem ntrval 21560
Description: The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
ntrval ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝐽 ∩ 𝒫 𝑆))

Proof of Theorem ntrval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . . 5 𝑋 = 𝐽
21ntrfval 21548 . . . 4 (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
32fveq1d 6668 . . 3 (𝐽 ∈ Top → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆))
43adantr 481 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆))
5 eqid 2825 . . 3 (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))
6 pweq 4544 . . . . 5 (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
76ineq2d 4192 . . . 4 (𝑥 = 𝑆 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
87unieqd 4846 . . 3 (𝑥 = 𝑆 (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
91topopn 21430 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
10 elpw2g 5243 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
119, 10syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1211biimpar 478 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
13 inex1g 5219 . . . . 5 (𝐽 ∈ Top → (𝐽 ∩ 𝒫 𝑆) ∈ V)
1413adantr 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
15 uniexg 7460 . . . 4 ((𝐽 ∩ 𝒫 𝑆) ∈ V → (𝐽 ∩ 𝒫 𝑆) ∈ V)
1614, 15syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
175, 8, 12, 16fvmptd3 6786 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
184, 17eqtrd 2860 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  Vcvv 3499  cin 3938  wss 3939  𝒫 cpw 4541   cuni 4836  cmpt 5142  cfv 6351  Topctop 21417  intcnt 21541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-top 21418  df-ntr 21544
This theorem is referenced by:  ntropn  21573  clsval2  21574  ntrss2  21581  ssntr  21582  isopn3  21590  ntreq0  21601
  Copyright terms: Public domain W3C validator