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Theorem ntrrn 42486
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 = βˆͺ 𝐽
ntrrn.i 𝐼 = (intβ€˜π½)
Assertion
Ref Expression
ntrrn (𝐽 ∈ Top β†’ ran 𝐼 βŠ† 𝐽)

Proof of Theorem ntrrn
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ntrrn.i . . 3 𝐼 = (intβ€˜π½)
21rneqi 5896 . 2 ran 𝐼 = ran (intβ€˜π½)
3 vpwex 5336 . . . . . . . 8 𝒫 𝑠 ∈ V
43inex2 5279 . . . . . . 7 (𝐽 ∩ 𝒫 𝑠) ∈ V
54uniex 7682 . . . . . 6 βˆͺ (𝐽 ∩ 𝒫 𝑠) ∈ V
65rgenw 3065 . . . . 5 βˆ€π‘  ∈ 𝒫 𝑋βˆͺ (𝐽 ∩ 𝒫 𝑠) ∈ V
7 nfcv 2904 . . . . . 6 Ⅎ𝑠𝒫 𝑋
87fnmptf 6641 . . . . 5 (βˆ€π‘  ∈ 𝒫 𝑋βˆͺ (𝐽 ∩ 𝒫 𝑠) ∈ V β†’ (𝑠 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)
96, 8mp1i 13 . . . 4 (𝐽 ∈ Top β†’ (𝑠 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋)
10 ntrrn.x . . . . . 6 𝑋 = βˆͺ 𝐽
1110ntrfval 22398 . . . . 5 (𝐽 ∈ Top β†’ (intβ€˜π½) = (𝑠 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑠)))
1211fneq1d 6599 . . . 4 (𝐽 ∈ Top β†’ ((intβ€˜π½) Fn 𝒫 𝑋 ↔ (𝑠 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑠)) Fn 𝒫 𝑋))
139, 12mpbird 257 . . 3 (𝐽 ∈ Top β†’ (intβ€˜π½) Fn 𝒫 𝑋)
14 elpwi 4571 . . . . 5 (𝑠 ∈ 𝒫 𝑋 β†’ 𝑠 βŠ† 𝑋)
1510ntropn 22423 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑠 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘ ) ∈ 𝐽)
1615ex 414 . . . . 5 (𝐽 ∈ Top β†’ (𝑠 βŠ† 𝑋 β†’ ((intβ€˜π½)β€˜π‘ ) ∈ 𝐽))
1714, 16syl5 34 . . . 4 (𝐽 ∈ Top β†’ (𝑠 ∈ 𝒫 𝑋 β†’ ((intβ€˜π½)β€˜π‘ ) ∈ 𝐽))
1817ralrimiv 3139 . . 3 (𝐽 ∈ Top β†’ βˆ€π‘  ∈ 𝒫 𝑋((intβ€˜π½)β€˜π‘ ) ∈ 𝐽)
19 fnfvrnss 7072 . . 3 (((intβ€˜π½) Fn 𝒫 𝑋 ∧ βˆ€π‘  ∈ 𝒫 𝑋((intβ€˜π½)β€˜π‘ ) ∈ 𝐽) β†’ ran (intβ€˜π½) βŠ† 𝐽)
2013, 18, 19syl2anc 585 . 2 (𝐽 ∈ Top β†’ ran (intβ€˜π½) βŠ† 𝐽)
212, 20eqsstrid 3996 1 (𝐽 ∈ Top β†’ ran 𝐼 βŠ† 𝐽)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3447   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  βˆͺ cuni 4869   ↦ cmpt 5192  ran crn 5638   Fn wfn 6495  β€˜cfv 6500  Topctop 22265  intcnt 22391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22266  df-ntr 22394
This theorem is referenced by:  ntrf  42487
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