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Theorem ntrrn 39190
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 5553 . 2 ran 𝐼 = ran (int‘𝐽)
3 vpwex 5045 . . . . . . . 8 𝒫 𝑠 ∈ V
43inex2 4993 . . . . . . 7 (𝐽 ∩ 𝒫 𝑠) ∈ V
54uniex 7185 . . . . . 6 (𝐽 ∩ 𝒫 𝑠) ∈ V
65rgenw 3103 . . . . 5 𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠) ∈ V
7 nfcv 2939 . . . . . 6 𝑠𝒫 𝑋
87fnmptf 6225 . . . . 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 21154 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
1211fneq1d 6190 . . . 4 (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
139, 12mpbird 249 . . 3 (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋)
14 elpwi 4357 . . . . 5 (𝑠 ∈ 𝒫 𝑋𝑠𝑋)
1510ntropn 21179 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑠𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽)
1615ex 402 . . . . 5 (𝐽 ∈ Top → (𝑠𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
1714, 16syl5 34 . . . 4 (𝐽 ∈ Top → (𝑠 ∈ 𝒫 𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
1817ralrimiv 3144 . . 3 (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽)
19 fnfvrnss 6614 . . 3 (((int‘𝐽) Fn 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) → ran (int‘𝐽) ⊆ 𝐽)
2013, 18, 19syl2anc 580 . 2 (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽)
212, 20syl5eqss 3843 1 (𝐽 ∈ Top → ran 𝐼𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  wral 3087  Vcvv 3383  cin 3766  wss 3767  𝒫 cpw 4347   cuni 4626  cmpt 4920  ran crn 5311   Fn wfn 6094  cfv 6099  Topctop 21023  intcnt 21147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-top 21024  df-ntr 21150
This theorem is referenced by:  ntrf  39191
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