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Theorem dssmapclsntr 41206
 Description: The closure and interior operators on a topology are duals of each other. See also kur14lem2 32686. (Contributed by RP, 22-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
dssmapclsntr (𝐽 ∈ Top → 𝐾 = (𝐷𝐼))
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 dssmapclsntr
StepHypRef Expression
1 dssmapclsntr.x . . . . 5 𝑋 = 𝐽
2 dssmapclsntr.k . . . . 5 𝐾 = (cls‘𝐽)
3 dssmapclsntr.i . . . . 5 𝐼 = (int‘𝐽)
4 dssmapclsntr.o . . . . 5 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
5 dssmapclsntr.d . . . . 5 𝐷 = (𝑂𝑋)
61, 2, 3, 4, 5dssmapntrcls 41205 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝐷𝐾))
76eqcomd 2765 . . 3 (𝐽 ∈ Top → (𝐷𝐾) = 𝐼)
81topopn 21607 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
94, 5, 8dssmapf1od 41096 . . . 4 (𝐽 ∈ Top → 𝐷:(𝒫 𝑋m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋m 𝒫 𝑋))
101, 2clselmap 41204 . . . 4 (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋m 𝒫 𝑋))
11 f1ocnvfv 7028 . . . 4 ((𝐷:(𝒫 𝑋m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋m 𝒫 𝑋) ∧ 𝐾 ∈ (𝒫 𝑋m 𝒫 𝑋)) → ((𝐷𝐾) = 𝐼 → (𝐷𝐼) = 𝐾))
129, 10, 11syl2anc 588 . . 3 (𝐽 ∈ Top → ((𝐷𝐾) = 𝐼 → (𝐷𝐼) = 𝐾))
137, 12mpd 15 . 2 (𝐽 ∈ Top → (𝐷𝐼) = 𝐾)
144, 5, 8dssmapnvod 41095 . . 3 (𝐽 ∈ Top → 𝐷 = 𝐷)
1514fveq1d 6661 . 2 (𝐽 ∈ Top → (𝐷𝐼) = (𝐷𝐼))
1613, 15eqtr3d 2796 1 (𝐽 ∈ Top → 𝐾 = (𝐷𝐼))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112  Vcvv 3410   ∖ cdif 3856  𝒫 cpw 4495  ∪ cuni 4799   ↦ cmpt 5113  ◡ccnv 5524  –1-1-onto→wf1o 6335  ‘cfv 6336  (class class class)co 7151   ↑m cmap 8417  Topctop 21594  intcnt 21718  clsccl 21719 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7694  df-2nd 7695  df-map 8419  df-top 21595  df-cld 21720  df-ntr 21721  df-cls 21722 This theorem is referenced by: (None)
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