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Theorem ntrf 41733
Description: The interior function of a topology is a map from the powerset of the base set to the open sets of the topology. (Contributed by RP, 22-Apr-2021.)
Hypotheses
Ref Expression
ntrrn.x 𝑋 = 𝐽
ntrrn.i 𝐼 = (int‘𝐽)
Assertion
Ref Expression
ntrf (𝐽 ∈ Top → 𝐼:𝒫 𝑋𝐽)

Proof of Theorem ntrf
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 vpwex 5300 . . . . . 6 𝒫 𝑠 ∈ V
21inex2 5242 . . . . 5 (𝐽 ∩ 𝒫 𝑠) ∈ V
32uniex 7594 . . . 4 (𝐽 ∩ 𝒫 𝑠) ∈ V
4 eqid 2738 . . . 4 (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠))
53, 4fnmpti 6576 . . 3 (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋
6 ntrrn.i . . . . 5 𝐼 = (int‘𝐽)
7 ntrrn.x . . . . . 6 𝑋 = 𝐽
87ntrfval 22175 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
96, 8eqtrid 2790 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
109fneq1d 6526 . . 3 (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
115, 10mpbiri 257 . 2 (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋)
127, 6ntrrn 41732 . 2 (𝐽 ∈ Top → ran 𝐼𝐽)
13 df-f 6437 . 2 (𝐼:𝒫 𝑋𝐽 ↔ (𝐼 Fn 𝒫 𝑋 ∧ ran 𝐼𝐽))
1411, 12, 13sylanbrc 583 1 (𝐽 ∈ Top → 𝐼:𝒫 𝑋𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  cin 3886  wss 3887  𝒫 cpw 4533   cuni 4839  cmpt 5157  ran crn 5590   Fn wfn 6428  wf 6429  cfv 6433  Topctop 22042  intcnt 22168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-top 22043  df-ntr 22171
This theorem is referenced by:  ntrf2  41734
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