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Theorem dssmapntrcls 37905
Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 30894. (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 4809 . . . . . . 7 𝒫 𝑡 ∈ V
21inex2 4760 . . . . . 6 (𝐽 ∩ 𝒫 𝑡) ∈ V
32uniex 6906 . . . . 5 (𝐽 ∩ 𝒫 𝑡) ∈ V
43rgenw 2919 . . . 4 𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡) ∈ V
5 nfcv 2761 . . . . 5 𝑡𝒫 𝑋
65fnmptf 5973 . . . 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 20738 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)))
118, 10syl5eq 2667 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)))
1211fneq1d 5939 . . 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 20636 . . . . . 6 (𝐽 ∈ Top → 𝑋𝐽)
1714, 15, 16dssmapf1od 37794 . . . . 5 (𝐽 ∈ Top → 𝐷:(𝒫 𝑋𝑚 𝒫 𝑋)–1-1-onto→(𝒫 𝑋𝑚 𝒫 𝑋))
18 f1of 6094 . . . . 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 37904 . . . 4 (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋𝑚 𝒫 𝑋))
2219, 21ffvelrnd 6316 . . 3 (𝐽 ∈ Top → (𝐷𝐾) ∈ (𝒫 𝑋𝑚 𝒫 𝑋))
23 elmapfn 7824 . . 3 ((𝐷𝐾) ∈ (𝒫 𝑋𝑚 𝒫 𝑋) → (𝐷𝐾) Fn 𝒫 𝑋)
2422, 23syl 17 . 2 (𝐽 ∈ Top → (𝐷𝐾) Fn 𝒫 𝑋)
25 elpwi 4140 . . . . 5 (𝑡 ∈ 𝒫 𝑋𝑡𝑋)
269ntrval2 20765 . . . . 5 ((𝐽 ∈ Top ∧ 𝑡𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡))))
2725, 26sylan2 491 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡))))
288fveq1i 6149 . . . 4 (𝐼𝑡) = ((int‘𝐽)‘𝑡)
2920fveq1i 6149 . . . . 5 (𝐾‘(𝑋𝑡)) = ((cls‘𝐽)‘(𝑋𝑡))
3029difeq2i 3703 . . . 4 (𝑋 ∖ (𝐾‘(𝑋𝑡))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡)))
3127, 28, 303eqtr4g 2680 . . 3 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼𝑡) = (𝑋 ∖ (𝐾‘(𝑋𝑡))))
3216adantr 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑋𝐽)
3321adantr 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝐾 ∈ (𝒫 𝑋𝑚 𝒫 𝑋))
34 eqid 2621 . . . 4 (𝐷𝐾) = (𝐷𝐾)
35 simpr 477 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑡 ∈ 𝒫 𝑋)
36 eqid 2621 . . . 4 ((𝐷𝐾)‘𝑡) = ((𝐷𝐾)‘𝑡)
3714, 15, 32, 33, 34, 35, 36dssmapfv3d 37792 . . 3 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((𝐷𝐾)‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋𝑡))))
3831, 37eqtr4d 2658 . 2 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼𝑡) = ((𝐷𝐾)‘𝑡))
3913, 24, 38eqfnfvd 6270 1 (𝐽 ∈ Top → 𝐼 = (𝐷𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402  cmpt 4673   Fn wfn 5842  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  Topctop 20617  intcnt 20731  clsccl 20732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-top 20621  df-cld 20733  df-ntr 20734  df-cls 20735
This theorem is referenced by:  dssmapclsntr  37906
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