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Theorem ntrrn 40996
 Description: The range of the interior function of a topology a subset of 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
ntrrn (𝐽 ∈ Top → ran 𝐼𝐽)

Proof of Theorem ntrrn
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ntrrn.i . . 3 𝐼 = (int‘𝐽)
21rneqi 5777 . 2 ran 𝐼 = ran (int‘𝐽)
3 vpwex 5247 . . . . . . . 8 𝒫 𝑠 ∈ V
43inex2 5190 . . . . . . 7 (𝐽 ∩ 𝒫 𝑠) ∈ V
54uniex 7460 . . . . . 6 (𝐽 ∩ 𝒫 𝑠) ∈ V
65rgenw 3118 . . . . 5 𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠) ∈ V
7 nfcv 2955 . . . . . 6 𝑠𝒫 𝑋
87fnmptf 6464 . . . . 5 (∀𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠) ∈ V → (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)
96, 8mp1i 13 . . . 4 (𝐽 ∈ Top → (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)
10 ntrrn.x . . . . . 6 𝑋 = 𝐽
1110ntrfval 21670 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
1211fneq1d 6424 . . . 4 (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
139, 12mpbird 260 . . 3 (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋)
14 elpwi 4509 . . . . 5 (𝑠 ∈ 𝒫 𝑋𝑠𝑋)
1510ntropn 21695 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑠𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽)
1615ex 416 . . . . 5 (𝐽 ∈ Top → (𝑠𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
1714, 16syl5 34 . . . 4 (𝐽 ∈ Top → (𝑠 ∈ 𝒫 𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
1817ralrimiv 3148 . . 3 (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽)
19 fnfvrnss 6871 . . 3 (((int‘𝐽) Fn 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) → ran (int‘𝐽) ⊆ 𝐽)
2013, 18, 19syl2anc 587 . 2 (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽)
212, 20eqsstrid 3965 1 (𝐽 ∈ Top → ran 𝐼𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3442   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4500  ∪ cuni 4804   ↦ cmpt 5114  ran crn 5524   Fn wfn 6327  ‘cfv 6332  Topctop 21539  intcnt 21663 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-top 21540  df-ntr 21666 This theorem is referenced by:  ntrf  40997
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