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Theorem ntrval 20598
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 20586 . . . 4 (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
32fveq1d 6090 . . 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 20484 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
6 elpw2g 4749 . . . . 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 4724 . . . . 5 (𝐽 ∈ Top → (𝐽 ∩ 𝒫 𝑆) ∈ V)
109adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
11 uniexg 6831 . . . 4 ((𝐽 ∩ 𝒫 𝑆) ∈ V → (𝐽 ∩ 𝒫 𝑆) ∈ V)
1210, 11syl 17 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
13 pweq 4111 . . . . . 6 (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
1413ineq2d 3776 . . . . 5 (𝑥 = 𝑆 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
1514unieqd 4377 . . . 4 (𝑥 = 𝑆 (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
16 eqid 2610 . . . 4 (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))
1715, 16fvmptg 6174 . . 3 ((𝑆 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑆) ∈ V) → ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
188, 12, 17syl2anc 691 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
194, 18eqtrd 2644 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cin 3539  wss 3540  𝒫 cpw 4108   cuni 4367  cmpt 4638  cfv 5790  Topctop 20465  intcnt 20579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-top 20469  df-ntr 20582
This theorem is referenced by:  ntropn  20611  clsval2  20612  ntrss2  20619  ssntr  20620  isopn3  20628  ntreq0  20639
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