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Theorem ntrrn 40350
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 5800 . 2 ran 𝐼 = ran (int‘𝐽)
3 vpwex 5269 . . . . . . . 8 𝒫 𝑠 ∈ V
43inex2 5213 . . . . . . 7 (𝐽 ∩ 𝒫 𝑠) ∈ V
54uniex 7454 . . . . . 6 (𝐽 ∩ 𝒫 𝑠) ∈ V
65rgenw 3147 . . . . 5 𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠) ∈ V
7 nfcv 2974 . . . . . 6 𝑠𝒫 𝑋
87fnmptf 6477 . . . . 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 21560 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
1211fneq1d 6439 . . . 4 (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
139, 12mpbird 258 . . 3 (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋)
14 elpwi 4547 . . . . 5 (𝑠 ∈ 𝒫 𝑋𝑠𝑋)
1510ntropn 21585 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑠𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽)
1615ex 413 . . . . 5 (𝐽 ∈ Top → (𝑠𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
1714, 16syl5 34 . . . 4 (𝐽 ∈ Top → (𝑠 ∈ 𝒫 𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
1817ralrimiv 3178 . . 3 (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽)
19 fnfvrnss 6876 . . 3 (((int‘𝐽) Fn 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) → ran (int‘𝐽) ⊆ 𝐽)
2013, 18, 19syl2anc 584 . 2 (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽)
212, 20eqsstrid 4012 1 (𝐽 ∈ Top → ran 𝐼𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830  cmpt 5137  ran crn 5549   Fn wfn 6343  cfv 6348  Topctop 21429  intcnt 21553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21430  df-ntr 21556
This theorem is referenced by:  ntrf  40351
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