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Theorem ntrf 37441
 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 4775 . . . . . 6 𝒫 𝑠 ∈ V
21inex2 4728 . . . . 5 (𝐽 ∩ 𝒫 𝑠) ∈ V
32uniex 6851 . . . 4 (𝐽 ∩ 𝒫 𝑠) ∈ V
4 eqid 2610 . . . 4 (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠))
53, 4fnmpti 5935 . . 3 (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋
6 ntrrn.i . . . . 5 𝐼 = (int‘𝐽)
7 ntrrn.x . . . . . 6 𝑋 = 𝐽
87ntrfval 20638 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
96, 8syl5eq 2656 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
109fneq1d 5895 . . 3 (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
115, 10mpbiri 247 . 2 (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋)
127, 6ntrrn 37440 . 2 (𝐽 ∈ Top → ran 𝐼𝐽)
13 df-f 5808 . 2 (𝐼:𝒫 𝑋𝐽 ↔ (𝐼 Fn 𝒫 𝑋 ∧ ran 𝐼𝐽))
1411, 12, 13sylanbrc 695 1 (𝐽 ∈ Top → 𝐼:𝒫 𝑋𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   ↦ cmpt 4643  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  Topctop 20517  intcnt 20631 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 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 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 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-ntr 20634 This theorem is referenced by:  ntrf2  37442
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