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Theorem ntrrn 42375
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 5892 . 2 ran 𝐼 = ran (int‘𝐽)
3 vpwex 5332 . . . . . . . 8 𝒫 𝑠 ∈ V
43inex2 5275 . . . . . . 7 (𝐽 ∩ 𝒫 𝑠) ∈ V
54uniex 7677 . . . . . 6 (𝐽 ∩ 𝒫 𝑠) ∈ V
65rgenw 3068 . . . . 5 𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠) ∈ V
7 nfcv 2907 . . . . . 6 𝑠𝒫 𝑋
87fnmptf 6637 . . . . 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 22373 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)))
1211fneq1d 6595 . . . 4 (𝐽 ∈ Top → ((int‘𝐽) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
139, 12mpbird 256 . . 3 (𝐽 ∈ Top → (int‘𝐽) Fn 𝒫 𝑋)
14 elpwi 4567 . . . . 5 (𝑠 ∈ 𝒫 𝑋𝑠𝑋)
1510ntropn 22398 . . . . . 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 3142 . . 3 (𝐽 ∈ Top → ∀𝑠 ∈ 𝒫 𝑋((int‘𝐽)‘𝑠) ∈ 𝐽)
19 fnfvrnss 7067 . . 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 3992 1 (𝐽 ∈ Top → ran 𝐼𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cin 3909  wss 3910  𝒫 cpw 4560   cuni 4865  cmpt 5188  ran crn 5634   Fn wfn 6491  cfv 6496  Topctop 22240  intcnt 22366
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-top 22241  df-ntr 22369
This theorem is referenced by:  ntrf  42376
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