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Theorem ntrf 43695
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 5319 . . . . 5 (𝐽 ∩ 𝒫 𝑠) ∈ V
32uniex 7747 . . . 4 (𝐽 ∩ 𝒫 𝑠) ∈ V
4 eqid 2725 . . . 4 (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠))
53, 4fnmpti 6699 . . 3 (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋
6 ntrrn.i . . . . 5 𝐼 = (int‘𝐽)
7 ntrrn.x . . . . . 6 𝑋 = 𝐽
87ntrfval 22972 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
96, 8eqtrid 2777 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
109fneq1d 6648 . . 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 43694 . 2 (𝐽 ∈ Top → ran 𝐼𝐽)
13 df-f 6553 . 2 (𝐼:𝒫 𝑋𝐽 ↔ (𝐼 Fn 𝒫 𝑋 ∧ ran 𝐼𝐽))
1411, 12, 13sylanbrc 581 1 (𝐽 ∈ Top → 𝐼:𝒫 𝑋𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cin 3943  wss 3944  𝒫 cpw 4604   cuni 4909  cmpt 5232  ran crn 5679   Fn wfn 6544  wf 6545  cfv 6549  Topctop 22839  intcnt 22965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-top 22840  df-ntr 22968
This theorem is referenced by:  ntrf2  43696
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