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Theorem dssmapntrcls 41002
 Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 32633. (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 5247 . . . . . . 7 𝒫 𝑡 ∈ V
21inex2 5190 . . . . . 6 (𝐽 ∩ 𝒫 𝑡) ∈ V
32uniex 7460 . . . . 5 (𝐽 ∩ 𝒫 𝑡) ∈ V
43rgenw 3118 . . . 4 𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡) ∈ V
5 nfcv 2955 . . . . 5 𝑡𝒫 𝑋
65fnmptf 6464 . . . 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 21670 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)))
118, 10syl5eq 2845 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)))
1211fneq1d 6424 . . 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 21552 . . . . . 6 (𝐽 ∈ Top → 𝑋𝐽)
1714, 15, 16dssmapf1od 40893 . . . . 5 (𝐽 ∈ Top → 𝐷:(𝒫 𝑋m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋m 𝒫 𝑋))
18 f1of 6599 . . . . 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 41001 . . . 4 (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋m 𝒫 𝑋))
2219, 21ffvelrnd 6839 . . 3 (𝐽 ∈ Top → (𝐷𝐾) ∈ (𝒫 𝑋m 𝒫 𝑋))
23 elmapfn 8430 . . 3 ((𝐷𝐾) ∈ (𝒫 𝑋m 𝒫 𝑋) → (𝐷𝐾) Fn 𝒫 𝑋)
2422, 23syl 17 . 2 (𝐽 ∈ Top → (𝐷𝐾) Fn 𝒫 𝑋)
25 elpwi 4509 . . . . 5 (𝑡 ∈ 𝒫 𝑋𝑡𝑋)
269ntrval2 21697 . . . . 5 ((𝐽 ∈ Top ∧ 𝑡𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡))))
2725, 26sylan2 595 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡))))
288fveq1i 6656 . . . 4 (𝐼𝑡) = ((int‘𝐽)‘𝑡)
2920fveq1i 6656 . . . . 5 (𝐾‘(𝑋𝑡)) = ((cls‘𝐽)‘(𝑋𝑡))
3029difeq2i 4050 . . . 4 (𝑋 ∖ (𝐾‘(𝑋𝑡))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡)))
3127, 28, 303eqtr4g 2858 . . 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 2798 . . . 4 (𝐷𝐾) = (𝐷𝐾)
35 simpr 488 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑡 ∈ 𝒫 𝑋)
36 eqid 2798 . . . 4 ((𝐷𝐾)‘𝑡) = ((𝐷𝐾)‘𝑡)
3714, 15, 32, 33, 34, 35, 36dssmapfv3d 40891 . . 3 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((𝐷𝐾)‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋𝑡))))
3831, 37eqtr4d 2836 . 2 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼𝑡) = ((𝐷𝐾)‘𝑡))
3913, 24, 38eqfnfvd 6792 1 (𝐽 ∈ Top → 𝐼 = (𝐷𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3442   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4500  ∪ cuni 4804   ↦ cmpt 5114   Fn wfn 6327  ⟶wf 6328  –1-1-onto→wf1o 6331  ‘cfv 6332  (class class class)co 7145   ↑m cmap 8407  Topctop 21539  intcnt 21663  clsccl 21664 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-iin 4888  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7684  df-2nd 7685  df-map 8409  df-top 21540  df-cld 21665  df-ntr 21666  df-cls 21667 This theorem is referenced by:  dssmapclsntr  41003
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