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Theorem ntrrn 37899
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 5312 . 2 ran 𝐼 = ran (int‘𝐽)
3 vpwex 4809 . . . . . . . 8 𝒫 𝑠 ∈ V
43inex2 4760 . . . . . . 7 (𝐽 ∩ 𝒫 𝑠) ∈ V
54uniex 6906 . . . . . 6 (𝐽 ∩ 𝒫 𝑠) ∈ V
65rgenw 2919 . . . . 5 𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠) ∈ V
7 nfcv 2761 . . . . . 6 𝑠𝒫 𝑋
87fnmptf 5973 . . . . 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 20738 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
1211fneq1d 5939 . . . 4 (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
139, 12mpbird 247 . . 3 (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋)
14 elpwi 4140 . . . . 5 (𝑠 ∈ 𝒫 𝑋𝑠𝑋)
1510ntropn 20763 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑠𝑋) → ((int‘𝐽)‘𝑠) ∈ 𝐽)
1615ex 450 . . . . 5 (𝐽 ∈ Top → (𝑠𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
1714, 16syl5 34 . . . 4 (𝐽 ∈ Top → (𝑠 ∈ 𝒫 𝑋 → ((int‘𝐽)‘𝑠) ∈ 𝐽))
1817ralrimiv 2959 . . 3 (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽)
19 fnfvrnss 6345 . . 3 (((int‘𝐽) Fn 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽) → ran (int‘𝐽) ⊆ 𝐽)
2013, 18, 19syl2anc 692 . 2 (𝐽 ∈ Top → ran (int‘𝐽) ⊆ 𝐽)
212, 20syl5eqss 3628 1 (𝐽 ∈ Top → ran 𝐼𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402  cmpt 4673  ran crn 5075   Fn wfn 5842  cfv 5847  Topctop 20617  intcnt 20731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-top 20621  df-ntr 20734
This theorem is referenced by:  ntrf  37900
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