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Theorem ntrf 42557
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 5359 . . . . . 6 𝒫 𝑠 ∈ V
21inex2 5302 . . . . 5 (𝐽 ∩ 𝒫 𝑠) ∈ V
32uniex 7705 . . . 4 (𝐽 ∩ 𝒫 𝑠) ∈ V
4 eqid 2731 . . . 4 (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠))
53, 4fnmpti 6671 . . 3 (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋
6 ntrrn.i . . . . 5 𝐼 = (int‘𝐽)
7 ntrrn.x . . . . . 6 𝑋 = 𝐽
87ntrfval 22434 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
96, 8eqtrid 2783 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
109fneq1d 6622 . . 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 42556 . 2 (𝐽 ∈ Top → ran 𝐼𝐽)
13 df-f 6527 . 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 1541  wcel 2106  cin 3934  wss 3935  𝒫 cpw 4587   cuni 4892  cmpt 5215  ran crn 5661   Fn wfn 6518  wf 6519  cfv 6523  Topctop 22301  intcnt 22427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5269  ax-sep 5283  ax-nul 5290  ax-pow 5347  ax-pr 5411  ax-un 7699
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3372  df-rab 3426  df-v 3468  df-sbc 3765  df-csb 3881  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-nul 4310  df-if 4514  df-pw 4589  df-sn 4614  df-pr 4616  df-op 4620  df-uni 4893  df-iun 4983  df-br 5133  df-opab 5195  df-mpt 5216  df-id 5558  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6475  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-top 22302  df-ntr 22430
This theorem is referenced by:  ntrf2  42558
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