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Theorem ntrf 42487
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 = βˆͺ 𝐽
ntrrn.i 𝐼 = (intβ€˜π½)
Assertion
Ref Expression
ntrf (𝐽 ∈ Top β†’ 𝐼:𝒫 π‘‹βŸΆπ½)

Proof of Theorem ntrf
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 vpwex 5336 . . . . . 6 𝒫 𝑠 ∈ V
21inex2 5279 . . . . 5 (𝐽 ∩ 𝒫 𝑠) ∈ V
32uniex 7682 . . . 4 βˆͺ (𝐽 ∩ 𝒫 𝑠) ∈ V
4 eqid 2733 . . . 4 (𝑠 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑠)) = (𝑠 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑠))
53, 4fnmpti 6648 . . 3 (𝑠 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋
6 ntrrn.i . . . . 5 𝐼 = (intβ€˜π½)
7 ntrrn.x . . . . . 6 𝑋 = βˆͺ 𝐽
87ntrfval 22398 . . . . 5 (𝐽 ∈ Top β†’ (intβ€˜π½) = (𝑠 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑠)))
96, 8eqtrid 2785 . . . 4 (𝐽 ∈ Top β†’ 𝐼 = (𝑠 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑠)))
109fneq1d 6599 . . 3 (𝐽 ∈ Top β†’ (𝐼 Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
115, 10mpbiri 258 . 2 (𝐽 ∈ Top β†’ 𝐼 Fn 𝒫 𝑋)
127, 6ntrrn 42486 . 2 (𝐽 ∈ Top β†’ ran 𝐼 βŠ† 𝐽)
13 df-f 6504 . 2 (𝐼:𝒫 π‘‹βŸΆπ½ ↔ (𝐼 Fn 𝒫 𝑋 ∧ ran 𝐼 βŠ† 𝐽))
1411, 12, 13sylanbrc 584 1 (𝐽 ∈ Top β†’ 𝐼:𝒫 π‘‹βŸΆπ½)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  βˆͺ cuni 4869   ↦ cmpt 5192  ran crn 5638   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  Topctop 22265  intcnt 22391
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22266  df-ntr 22394
This theorem is referenced by:  ntrf2  42488
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