Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dssmapntrcls Structured version   Visualization version   GIF version

Theorem dssmapntrcls 42864
Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 34186. (Contributed by RP, 21-Apr-2021.)
Hypotheses
Ref Expression
dssmapclsntr.x 𝑋 = βˆͺ 𝐽
dssmapclsntr.k 𝐾 = (clsβ€˜π½)
dssmapclsntr.i 𝐼 = (intβ€˜π½)
dssmapclsntr.o 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 βˆ– (π‘“β€˜(𝑏 βˆ– 𝑠))))))
dssmapclsntr.d 𝐷 = (π‘‚β€˜π‘‹)
Assertion
Ref Expression
dssmapntrcls (𝐽 ∈ Top β†’ 𝐼 = (π·β€˜πΎ))
Distinct variable groups:   𝐽,𝑏,𝑓,𝑠   𝑓,𝐾,𝑠   𝑋,𝑏,𝑓,𝑠
Allowed substitution hints:   𝐷(𝑓,𝑠,𝑏)   𝐼(𝑓,𝑠,𝑏)   𝐾(𝑏)   𝑂(𝑓,𝑠,𝑏)

Proof of Theorem dssmapntrcls
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 vpwex 5374 . . . . . . 7 𝒫 𝑑 ∈ V
21inex2 5317 . . . . . 6 (𝐽 ∩ 𝒫 𝑑) ∈ V
32uniex 7727 . . . . 5 βˆͺ (𝐽 ∩ 𝒫 𝑑) ∈ V
43rgenw 3065 . . . 4 βˆ€π‘‘ ∈ 𝒫 𝑋βˆͺ (𝐽 ∩ 𝒫 𝑑) ∈ V
5 nfcv 2903 . . . . 5 Ⅎ𝑑𝒫 𝑋
65fnmptf 6683 . . . 4 (βˆ€π‘‘ ∈ 𝒫 𝑋βˆͺ (𝐽 ∩ 𝒫 𝑑) ∈ V β†’ (𝑑 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑑)) Fn 𝒫 𝑋)
74, 6mp1i 13 . . 3 (𝐽 ∈ Top β†’ (𝑑 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑑)) Fn 𝒫 𝑋)
8 dssmapclsntr.i . . . . 5 𝐼 = (intβ€˜π½)
9 dssmapclsntr.x . . . . . 6 𝑋 = βˆͺ 𝐽
109ntrfval 22519 . . . . 5 (𝐽 ∈ Top β†’ (intβ€˜π½) = (𝑑 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑑)))
118, 10eqtrid 2784 . . . 4 (𝐽 ∈ Top β†’ 𝐼 = (𝑑 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑑)))
1211fneq1d 6639 . . 3 (𝐽 ∈ Top β†’ (𝐼 Fn 𝒫 𝑋 ↔ (𝑑 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑑)) Fn 𝒫 𝑋))
137, 12mpbird 256 . 2 (𝐽 ∈ Top β†’ 𝐼 Fn 𝒫 𝑋)
14 dssmapclsntr.o . . . . . 6 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 βˆ– (π‘“β€˜(𝑏 βˆ– 𝑠))))))
15 dssmapclsntr.d . . . . . 6 𝐷 = (π‘‚β€˜π‘‹)
169topopn 22399 . . . . . 6 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
1714, 15, 16dssmapf1od 42757 . . . . 5 (𝐽 ∈ Top β†’ 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-ontoβ†’(𝒫 𝑋 ↑m 𝒫 𝑋))
18 f1of 6830 . . . . 5 (𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-ontoβ†’(𝒫 𝑋 ↑m 𝒫 𝑋) β†’ 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)⟢(𝒫 𝑋 ↑m 𝒫 𝑋))
1917, 18syl 17 . . . 4 (𝐽 ∈ Top β†’ 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)⟢(𝒫 𝑋 ↑m 𝒫 𝑋))
20 dssmapclsntr.k . . . . 5 𝐾 = (clsβ€˜π½)
219, 20clselmap 42863 . . . 4 (𝐽 ∈ Top β†’ 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
2219, 21ffvelcdmd 7084 . . 3 (𝐽 ∈ Top β†’ (π·β€˜πΎ) ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
23 elmapfn 8855 . . 3 ((π·β€˜πΎ) ∈ (𝒫 𝑋 ↑m 𝒫 𝑋) β†’ (π·β€˜πΎ) Fn 𝒫 𝑋)
2422, 23syl 17 . 2 (𝐽 ∈ Top β†’ (π·β€˜πΎ) Fn 𝒫 𝑋)
25 elpwi 4608 . . . . 5 (𝑑 ∈ 𝒫 𝑋 β†’ 𝑑 βŠ† 𝑋)
269ntrval2 22546 . . . . 5 ((𝐽 ∈ Top ∧ 𝑑 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘‘) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑑))))
2725, 26sylan2 593 . . . 4 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ ((intβ€˜π½)β€˜π‘‘) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑑))))
288fveq1i 6889 . . . 4 (πΌβ€˜π‘‘) = ((intβ€˜π½)β€˜π‘‘)
2920fveq1i 6889 . . . . 5 (πΎβ€˜(𝑋 βˆ– 𝑑)) = ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑑))
3029difeq2i 4118 . . . 4 (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– 𝑑))) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑑)))
3127, 28, 303eqtr4g 2797 . . 3 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ (πΌβ€˜π‘‘) = (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– 𝑑))))
3216adantr 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ 𝑋 ∈ 𝐽)
3321adantr 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
34 eqid 2732 . . . 4 (π·β€˜πΎ) = (π·β€˜πΎ)
35 simpr 485 . . . 4 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ 𝑑 ∈ 𝒫 𝑋)
36 eqid 2732 . . . 4 ((π·β€˜πΎ)β€˜π‘‘) = ((π·β€˜πΎ)β€˜π‘‘)
3714, 15, 32, 33, 34, 35, 36dssmapfv3d 42755 . . 3 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ ((π·β€˜πΎ)β€˜π‘‘) = (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– 𝑑))))
3831, 37eqtr4d 2775 . 2 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ (πΌβ€˜π‘‘) = ((π·β€˜πΎ)β€˜π‘‘))
3913, 24, 38eqfnfvd 7032 1 (𝐽 ∈ Top β†’ 𝐼 = (π·β€˜πΎ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907   ↦ cmpt 5230   Fn wfn 6535  βŸΆwf 6536  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405   ↑m cmap 8816  Topctop 22386  intcnt 22512  clsccl 22513
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-top 22387  df-cld 22514  df-ntr 22515  df-cls 22516
This theorem is referenced by:  dssmapclsntr  42865
  Copyright terms: Public domain W3C validator