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 38946
Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 31517. (Contributed by RP, 21-Apr-2021.)
Hypotheses
Ref Expression
dssmapclsntr.x 𝑋 = 𝐽
dssmapclsntr.k 𝐾 = (cls‘𝐽)
dssmapclsntr.i 𝐼 = (int‘𝐽)
dssmapclsntr.o 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
dssmapclsntr.d 𝐷 = (𝑂𝑋)
Assertion
Ref Expression
dssmapntrcls (𝐽 ∈ Top → 𝐼 = (𝐷𝐾))
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 𝑡 is distinct from all other variables.
StepHypRef Expression
1 vpwex 4998 . . . . . . 7 𝒫 𝑡 ∈ V
21inex2 4952 . . . . . 6 (𝐽 ∩ 𝒫 𝑡) ∈ V
32uniex 7119 . . . . 5 (𝐽 ∩ 𝒫 𝑡) ∈ V
43rgenw 3062 . . . 4 𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡) ∈ V
5 nfcv 2902 . . . . 5 𝑡𝒫 𝑋
65fnmptf 6177 . . . 4 (∀𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡) ∈ V → (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)) Fn 𝒫 𝑋)
74, 6mp1i 13 . . 3 (𝐽 ∈ Top → (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)) Fn 𝒫 𝑋)
8 dssmapclsntr.i . . . . 5 𝐼 = (int‘𝐽)
9 dssmapclsntr.x . . . . . 6 𝑋 = 𝐽
109ntrfval 21050 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)))
118, 10syl5eq 2806 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)))
1211fneq1d 6142 . . 3 (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)) Fn 𝒫 𝑋))
137, 12mpbird 247 . 2 (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋)
14 dssmapclsntr.o . . . . . 6 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
15 dssmapclsntr.d . . . . . 6 𝐷 = (𝑂𝑋)
169topopn 20933 . . . . . 6 (𝐽 ∈ Top → 𝑋𝐽)
1714, 15, 16dssmapf1od 38835 . . . . 5 (𝐽 ∈ Top → 𝐷:(𝒫 𝑋𝑚 𝒫 𝑋)–1-1-onto→(𝒫 𝑋𝑚 𝒫 𝑋))
18 f1of 6299 . . . . 5 (𝐷:(𝒫 𝑋𝑚 𝒫 𝑋)–1-1-onto→(𝒫 𝑋𝑚 𝒫 𝑋) → 𝐷:(𝒫 𝑋𝑚 𝒫 𝑋)⟶(𝒫 𝑋𝑚 𝒫 𝑋))
1917, 18syl 17 . . . 4 (𝐽 ∈ Top → 𝐷:(𝒫 𝑋𝑚 𝒫 𝑋)⟶(𝒫 𝑋𝑚 𝒫 𝑋))
20 dssmapclsntr.k . . . . 5 𝐾 = (cls‘𝐽)
219, 20clselmap 38945 . . . 4 (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋𝑚 𝒫 𝑋))
2219, 21ffvelrnd 6524 . . 3 (𝐽 ∈ Top → (𝐷𝐾) ∈ (𝒫 𝑋𝑚 𝒫 𝑋))
23 elmapfn 8048 . . 3 ((𝐷𝐾) ∈ (𝒫 𝑋𝑚 𝒫 𝑋) → (𝐷𝐾) Fn 𝒫 𝑋)
2422, 23syl 17 . 2 (𝐽 ∈ Top → (𝐷𝐾) Fn 𝒫 𝑋)
25 elpwi 4312 . . . . 5 (𝑡 ∈ 𝒫 𝑋𝑡𝑋)
269ntrval2 21077 . . . . 5 ((𝐽 ∈ Top ∧ 𝑡𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡))))
2725, 26sylan2 492 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡))))
288fveq1i 6354 . . . 4 (𝐼𝑡) = ((int‘𝐽)‘𝑡)
2920fveq1i 6354 . . . . 5 (𝐾‘(𝑋𝑡)) = ((cls‘𝐽)‘(𝑋𝑡))
3029difeq2i 3868 . . . 4 (𝑋 ∖ (𝐾‘(𝑋𝑡))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡)))
3127, 28, 303eqtr4g 2819 . . 3 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼𝑡) = (𝑋 ∖ (𝐾‘(𝑋𝑡))))
3216adantr 472 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑋𝐽)
3321adantr 472 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝐾 ∈ (𝒫 𝑋𝑚 𝒫 𝑋))
34 eqid 2760 . . . 4 (𝐷𝐾) = (𝐷𝐾)
35 simpr 479 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑡 ∈ 𝒫 𝑋)
36 eqid 2760 . . . 4 ((𝐷𝐾)‘𝑡) = ((𝐷𝐾)‘𝑡)
3714, 15, 32, 33, 34, 35, 36dssmapfv3d 38833 . . 3 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((𝐷𝐾)‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋𝑡))))
3831, 37eqtr4d 2797 . 2 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼𝑡) = ((𝐷𝐾)‘𝑡))
3913, 24, 38eqfnfvd 6478 1 (𝐽 ∈ Top → 𝐼 = (𝐷𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cdif 3712  cin 3714  wss 3715  𝒫 cpw 4302   cuni 4588  cmpt 4881   Fn wfn 6044  wf 6045  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6814  𝑚 cmap 8025  Topctop 20920  intcnt 21043  clsccl 21044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-top 20921  df-cld 21045  df-ntr 21046  df-cls 21047
This theorem is referenced by:  dssmapclsntr  38947
  Copyright terms: Public domain W3C validator