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Theorem dssmapntrcls 41415
Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 32882. (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 5270 . . . . . . 7 𝒫 𝑡 ∈ V
21inex2 5211 . . . . . 6 (𝐽 ∩ 𝒫 𝑡) ∈ V
32uniex 7529 . . . . 5 (𝐽 ∩ 𝒫 𝑡) ∈ V
43rgenw 3073 . . . 4 𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡) ∈ V
5 nfcv 2904 . . . . 5 𝑡𝒫 𝑋
65fnmptf 6514 . . . 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 21921 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)))
118, 10syl5eq 2790 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)))
1211fneq1d 6472 . . 3 (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)) Fn 𝒫 𝑋))
137, 12mpbird 260 . 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 21803 . . . . . 6 (𝐽 ∈ Top → 𝑋𝐽)
1714, 15, 16dssmapf1od 41306 . . . . 5 (𝐽 ∈ Top → 𝐷:(𝒫 𝑋m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋m 𝒫 𝑋))
18 f1of 6661 . . . . 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 41414 . . . 4 (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋m 𝒫 𝑋))
2219, 21ffvelrnd 6905 . . 3 (𝐽 ∈ Top → (𝐷𝐾) ∈ (𝒫 𝑋m 𝒫 𝑋))
23 elmapfn 8546 . . 3 ((𝐷𝐾) ∈ (𝒫 𝑋m 𝒫 𝑋) → (𝐷𝐾) Fn 𝒫 𝑋)
2422, 23syl 17 . 2 (𝐽 ∈ Top → (𝐷𝐾) Fn 𝒫 𝑋)
25 elpwi 4522 . . . . 5 (𝑡 ∈ 𝒫 𝑋𝑡𝑋)
269ntrval2 21948 . . . . 5 ((𝐽 ∈ Top ∧ 𝑡𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡))))
2725, 26sylan2 596 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡))))
288fveq1i 6718 . . . 4 (𝐼𝑡) = ((int‘𝐽)‘𝑡)
2920fveq1i 6718 . . . . 5 (𝐾‘(𝑋𝑡)) = ((cls‘𝐽)‘(𝑋𝑡))
3029difeq2i 4034 . . . 4 (𝑋 ∖ (𝐾‘(𝑋𝑡))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡)))
3127, 28, 303eqtr4g 2803 . . 3 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼𝑡) = (𝑋 ∖ (𝐾‘(𝑋𝑡))))
3216adantr 484 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑋𝐽)
3321adantr 484 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝐾 ∈ (𝒫 𝑋m 𝒫 𝑋))
34 eqid 2737 . . . 4 (𝐷𝐾) = (𝐷𝐾)
35 simpr 488 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑡 ∈ 𝒫 𝑋)
36 eqid 2737 . . . 4 ((𝐷𝐾)‘𝑡) = ((𝐷𝐾)‘𝑡)
3714, 15, 32, 33, 34, 35, 36dssmapfv3d 41304 . . 3 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((𝐷𝐾)‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋𝑡))))
3831, 37eqtr4d 2780 . 2 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼𝑡) = ((𝐷𝐾)‘𝑡))
3913, 24, 38eqfnfvd 6855 1 (𝐽 ∈ Top → 𝐼 = (𝐷𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wral 3061  Vcvv 3408  cdif 3863  cin 3865  wss 3866  𝒫 cpw 4513   cuni 4819  cmpt 5135   Fn wfn 6375  wf 6376  1-1-ontowf1o 6379  cfv 6380  (class class class)co 7213  m cmap 8508  Topctop 21790  intcnt 21914  clsccl 21915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-map 8510  df-top 21791  df-cld 21916  df-ntr 21917  df-cls 21918
This theorem is referenced by:  dssmapclsntr  41416
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