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Theorem ntrf 44136
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 5377 . . . . . 6 𝒫 𝑠 ∈ V
21inex2 5318 . . . . 5 (𝐽 ∩ 𝒫 𝑠) ∈ V
32uniex 7761 . . . 4 (𝐽 ∩ 𝒫 𝑠) ∈ V
4 eqid 2737 . . . 4 (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠))
53, 4fnmpti 6711 . . 3 (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋
6 ntrrn.i . . . . 5 𝐼 = (int‘𝐽)
7 ntrrn.x . . . . . 6 𝑋 = 𝐽
87ntrfval 23032 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
96, 8eqtrid 2789 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
109fneq1d 6661 . . 3 (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
115, 10mpbiri 258 . 2 (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋)
127, 6ntrrn 44135 . 2 (𝐽 ∈ Top → ran 𝐼𝐽)
13 df-f 6565 . 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 1540  wcel 2108  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907  cmpt 5225  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  Topctop 22899  intcnt 23025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-ntr 23028
This theorem is referenced by:  ntrf2  44137
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