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

Theorem ntrval 20888
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 20876 . . . 4 (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
32fveq1d 6231 . . 3 (𝐽 ∈ Top → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆))
43adantr 480 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆))
51topopn 20759 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
6 elpw2g 4857 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
75, 6syl 17 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
87biimpar 501 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
9 inex1g 4834 . . . . 5 (𝐽 ∈ Top → (𝐽 ∩ 𝒫 𝑆) ∈ V)
109adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
11 uniexg 6997 . . . 4 ((𝐽 ∩ 𝒫 𝑆) ∈ V → (𝐽 ∩ 𝒫 𝑆) ∈ V)
1210, 11syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
13 pweq 4194 . . . . . 6 (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
1413ineq2d 3847 . . . . 5 (𝑥 = 𝑆 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
1514unieqd 4478 . . . 4 (𝑥 = 𝑆 (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
16 eqid 2651 . . . 4 (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))
1715, 16fvmptg 6319 . . 3 ((𝑆 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑆) ∈ V) → ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
188, 12, 17syl2anc 694 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
194, 18eqtrd 2685 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468  cmpt 4762  cfv 5926  Topctop 20746  intcnt 20869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-top 20747  df-ntr 20872
This theorem is referenced by:  ntropn  20901  clsval2  20902  ntrss2  20909  ssntr  20910  isopn3  20918  ntreq0  20929
  Copyright terms: Public domain W3C validator