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Theorem ntrval 14278
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 14268 . . . 4 (𝐽 ∈ Top → (int‘𝐽) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)))
32fveq1d 5556 . . 3 (𝐽 ∈ Top → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆))
43adantr 276 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆))
5 eqid 2193 . . 3 (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥)) = (𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))
6 pweq 3604 . . . . 5 (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
76ineq2d 3360 . . . 4 (𝑥 = 𝑆 → (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
87unieqd 3846 . . 3 (𝑥 = 𝑆 (𝐽 ∩ 𝒫 𝑥) = (𝐽 ∩ 𝒫 𝑆))
91topopn 14176 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
10 elpw2g 4185 . . . . 5 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
119, 10syl 14 . . . 4 (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1211biimpar 297 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
13 inex1g 4165 . . . . 5 (𝐽 ∈ Top → (𝐽 ∩ 𝒫 𝑆) ∈ V)
1413adantr 276 . . . 4 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
15 uniexg 4470 . . . 4 ((𝐽 ∩ 𝒫 𝑆) ∈ V → (𝐽 ∩ 𝒫 𝑆) ∈ V)
1614, 15syl 14 . . 3 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝐽 ∩ 𝒫 𝑆) ∈ V)
175, 8, 12, 16fvmptd3 5651 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑥 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑥))‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
184, 17eqtrd 2226 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((int‘𝐽)‘𝑆) = (𝐽 ∩ 𝒫 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2164  Vcvv 2760  cin 3152  wss 3153  𝒫 cpw 3601   cuni 3835  cmpt 4090  cfv 5254  Topctop 14165  intcnt 14261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-top 14166  df-ntr 14264
This theorem is referenced by:  ntropn  14285  ntrss  14287  ntrss2  14289  ssntr  14290  isopn3  14293  ntreq0  14300
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