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Theorem ntrrn 43546
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 5933 . 2 ran 𝐼 = ran (intβ€˜π½)
3 vpwex 5371 . . . . . . . 8 𝒫 𝑠 ∈ V
43inex2 5312 . . . . . . 7 (𝐽 ∩ 𝒫 𝑠) ∈ V
54uniex 7740 . . . . . 6 βˆͺ (𝐽 ∩ 𝒫 𝑠) ∈ V
65rgenw 3061 . . . . 5 βˆ€π‘  ∈ 𝒫 𝑋βˆͺ (𝐽 ∩ 𝒫 𝑠) ∈ V
7 nfcv 2899 . . . . . 6 Ⅎ𝑠𝒫 𝑋
87fnmptf 6685 . . . . 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 22921 . . . . 5 (𝐽 ∈ Top β†’ (intβ€˜π½) = (𝑠 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑠)))
1211fneq1d 6641 . . . 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 4605 . . . . 5 (𝑠 ∈ 𝒫 𝑋 β†’ 𝑠 βŠ† 𝑋)
1510ntropn 22946 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑠 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘ ) ∈ 𝐽)
1615ex 412 . . . . 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 3141 . . 3 (𝐽 ∈ Top β†’ βˆ€π‘  ∈ 𝒫 𝑋((intβ€˜π½)β€˜π‘ ) ∈ 𝐽)
19 fnfvrnss 7125 . . 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 583 . 2 (𝐽 ∈ Top β†’ ran (intβ€˜π½) βŠ† 𝐽)
212, 20eqsstrid 4026 1 (𝐽 ∈ Top β†’ ran 𝐼 βŠ† 𝐽)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  βˆ€wral 3057  Vcvv 3470   ∩ cin 3944   βŠ† wss 3945  π’« cpw 4598  βˆͺ cuni 4903   ↦ cmpt 5225  ran crn 5673   Fn wfn 6537  β€˜cfv 6542  Topctop 22788  intcnt 22914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-top 22789  df-ntr 22917
This theorem is referenced by:  ntrf  43547
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