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Theorem dssmapclsntr 40357
Description: The closure and interior operators on a topology are duals of each other. See also kur14lem2 32351. (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 40356 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝐷𝐾))
76eqcomd 2824 . . 3 (𝐽 ∈ Top → (𝐷𝐾) = 𝐼)
81topopn 21442 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
94, 5, 8dssmapf1od 40245 . . . 4 (𝐽 ∈ Top → 𝐷:(𝒫 𝑋m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋m 𝒫 𝑋))
101, 2clselmap 40355 . . . 4 (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋m 𝒫 𝑋))
11 f1ocnvfv 7026 . . . 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 584 . . 3 (𝐽 ∈ Top → ((𝐷𝐾) = 𝐼 → (𝐷𝐼) = 𝐾))
137, 12mpd 15 . 2 (𝐽 ∈ Top → (𝐷𝐼) = 𝐾)
144, 5, 8dssmapnvod 40244 . . 3 (𝐽 ∈ Top → 𝐷 = 𝐷)
1514fveq1d 6665 . 2 (𝐽 ∈ Top → (𝐷𝐼) = (𝐷𝐼))
1613, 15eqtr3d 2855 1 (𝐽 ∈ Top → 𝐾 = (𝐷𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930  𝒫 cpw 4535   cuni 4830  cmpt 5137  ccnv 5547  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  m cmap 8395  Topctop 21429  intcnt 21553  clsccl 21554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-top 21430  df-cld 21555  df-ntr 21556  df-cls 21557
This theorem is referenced by: (None)
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