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Theorem ntrval 12316
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 12306 . . . 4 (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
32fveq1d 5430 . . 3 (𝐽 ∈ Top → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆))
43adantr 274 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆))
5 eqid 2140 . . 3 (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))
6 pweq 3517 . . . . 5 (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
76ineq2d 3281 . . . 4 (𝑥 = 𝑆 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
87unieqd 3754 . . 3 (𝑥 = 𝑆 (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
91topopn 12212 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
10 elpw2g 4088 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
119, 10syl 14 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1211biimpar 295 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
13 inex1g 4071 . . . . 5 (𝐽 ∈ Top → (𝐽 ∩ 𝒫 𝑆) ∈ V)
1413adantr 274 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
15 uniexg 4368 . . . 4 ((𝐽 ∩ 𝒫 𝑆) ∈ V → (𝐽 ∩ 𝒫 𝑆) ∈ V)
1614, 15syl 14 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
175, 8, 12, 16fvmptd3 5521 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
184, 17eqtrd 2173 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  Vcvv 2689  cin 3074  wss 3075  𝒫 cpw 3514   cuni 3743  cmpt 3996  cfv 5130  Topctop 12201  intcnt 12299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-top 12202  df-ntr 12302
This theorem is referenced by:  ntropn  12323  ntrss  12325  ntrss2  12327  ssntr  12328  isopn3  12331  ntreq0  12338
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