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Theorem dssmapntrcls 37446
Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 30443. (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 4775 . . . . . . 7 𝒫 𝑡 ∈ V
21inex2 4728 . . . . . 6 (𝐽 ∩ 𝒫 𝑡) ∈ V
32uniex 6851 . . . . 5 (𝐽 ∩ 𝒫 𝑡) ∈ V
43rgenw 2908 . . . 4 𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡) ∈ V
5 nfcv 2751 . . . . 5 𝑡𝒫 𝑋
65fnmptf 5929 . . . 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 20638 . . . . 5 (𝐽 ∈ Top → (int‘𝐽) = (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)))
118, 10syl5eq 2656 . . . 4 (𝐽 ∈ Top → 𝐼 = (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)))
1211fneq1d 5895 . . 3 (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑡 ∈ 𝒫 𝑋 (𝐽 ∩ 𝒫 𝑡)) Fn 𝒫 𝑋))
137, 12mpbird 246 . 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 20536 . . . . . 6 (𝐽 ∈ Top → 𝑋𝐽)
1714, 15, 16dssmapf1od 37335 . . . . 5 (𝐽 ∈ Top → 𝐷:(𝒫 𝑋𝑚 𝒫 𝑋)–1-1-onto→(𝒫 𝑋𝑚 𝒫 𝑋))
18 f1of 6050 . . . . 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 37445 . . . 4 (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋𝑚 𝒫 𝑋))
2219, 21ffvelrnd 6268 . . 3 (𝐽 ∈ Top → (𝐷𝐾) ∈ (𝒫 𝑋𝑚 𝒫 𝑋))
23 elmapfn 7766 . . 3 ((𝐷𝐾) ∈ (𝒫 𝑋𝑚 𝒫 𝑋) → (𝐷𝐾) Fn 𝒫 𝑋)
2422, 23syl 17 . 2 (𝐽 ∈ Top → (𝐷𝐾) Fn 𝒫 𝑋)
25 elpwi 4117 . . . . 5 (𝑡 ∈ 𝒫 𝑋𝑡𝑋)
269ntrval2 20665 . . . . 5 ((𝐽 ∈ Top ∧ 𝑡𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡))))
2725, 26sylan2 490 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡))))
288fveq1i 6104 . . . 4 (𝐼𝑡) = ((int‘𝐽)‘𝑡)
2920fveq1i 6104 . . . . 5 (𝐾‘(𝑋𝑡)) = ((cls‘𝐽)‘(𝑋𝑡))
3029difeq2i 3687 . . . 4 (𝑋 ∖ (𝐾‘(𝑋𝑡))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋𝑡)))
3127, 28, 303eqtr4g 2669 . . 3 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼𝑡) = (𝑋 ∖ (𝐾‘(𝑋𝑡))))
3216adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑋𝐽)
3321adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝐾 ∈ (𝒫 𝑋𝑚 𝒫 𝑋))
34 eqid 2610 . . . 4 (𝐷𝐾) = (𝐷𝐾)
35 simpr 476 . . . 4 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑡 ∈ 𝒫 𝑋)
36 eqid 2610 . . . 4 ((𝐷𝐾)‘𝑡) = ((𝐷𝐾)‘𝑡)
3714, 15, 32, 33, 34, 35, 36dssmapfv3d 37333 . . 3 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((𝐷𝐾)‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋𝑡))))
3831, 37eqtr4d 2647 . 2 ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼𝑡) = ((𝐷𝐾)‘𝑡))
3913, 24, 38eqfnfvd 6222 1 (𝐽 ∈ Top → 𝐼 = (𝐷𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cdif 3537  cin 3539  wss 3540  𝒫 cpw 4108   cuni 4372  cmpt 4643   Fn wfn 5799  wf 5800  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  𝑚 cmap 7744  Topctop 20517  intcnt 20631  clsccl 20632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-top 20521  df-cld 20633  df-ntr 20634  df-cls 20635
This theorem is referenced by:  dssmapclsntr  37447
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