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Theorem dssmapntrcls 43481
Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 34753. (Contributed by RP, 21-Apr-2021.)
Hypotheses
Ref Expression
dssmapclsntr.x 𝑋 = βˆͺ 𝐽
dssmapclsntr.k 𝐾 = (clsβ€˜π½)
dssmapclsntr.i 𝐼 = (intβ€˜π½)
dssmapclsntr.o 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 βˆ– (π‘“β€˜(𝑏 βˆ– 𝑠))))))
dssmapclsntr.d 𝐷 = (π‘‚β€˜π‘‹)
Assertion
Ref Expression
dssmapntrcls (𝐽 ∈ Top β†’ 𝐼 = (π·β€˜πΎ))
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 𝑑 is distinct from all other variables.
StepHypRef Expression
1 vpwex 5371 . . . . . . 7 𝒫 𝑑 ∈ V
21inex2 5312 . . . . . 6 (𝐽 ∩ 𝒫 𝑑) ∈ V
32uniex 7740 . . . . 5 βˆͺ (𝐽 ∩ 𝒫 𝑑) ∈ V
43rgenw 3060 . . . 4 βˆ€π‘‘ ∈ 𝒫 𝑋βˆͺ (𝐽 ∩ 𝒫 𝑑) ∈ V
5 nfcv 2898 . . . . 5 Ⅎ𝑑𝒫 𝑋
65fnmptf 6685 . . . 4 (βˆ€π‘‘ ∈ 𝒫 𝑋βˆͺ (𝐽 ∩ 𝒫 𝑑) ∈ V β†’ (𝑑 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑑)) Fn 𝒫 𝑋)
74, 6mp1i 13 . . 3 (𝐽 ∈ Top β†’ (𝑑 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑑)) Fn 𝒫 𝑋)
8 dssmapclsntr.i . . . . 5 𝐼 = (intβ€˜π½)
9 dssmapclsntr.x . . . . . 6 𝑋 = βˆͺ 𝐽
109ntrfval 22915 . . . . 5 (𝐽 ∈ Top β†’ (intβ€˜π½) = (𝑑 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑑)))
118, 10eqtrid 2779 . . . 4 (𝐽 ∈ Top β†’ 𝐼 = (𝑑 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑑)))
1211fneq1d 6641 . . 3 (𝐽 ∈ Top β†’ (𝐼 Fn 𝒫 𝑋 ↔ (𝑑 ∈ 𝒫 𝑋 ↦ βˆͺ (𝐽 ∩ 𝒫 𝑑)) Fn 𝒫 𝑋))
137, 12mpbird 257 . 2 (𝐽 ∈ Top β†’ 𝐼 Fn 𝒫 𝑋)
14 dssmapclsntr.o . . . . . 6 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 βˆ– (π‘“β€˜(𝑏 βˆ– 𝑠))))))
15 dssmapclsntr.d . . . . . 6 𝐷 = (π‘‚β€˜π‘‹)
169topopn 22795 . . . . . 6 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
1714, 15, 16dssmapf1od 43374 . . . . 5 (𝐽 ∈ Top β†’ 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-ontoβ†’(𝒫 𝑋 ↑m 𝒫 𝑋))
18 f1of 6833 . . . . 5 (𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-ontoβ†’(𝒫 𝑋 ↑m 𝒫 𝑋) β†’ 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)⟢(𝒫 𝑋 ↑m 𝒫 𝑋))
1917, 18syl 17 . . . 4 (𝐽 ∈ Top β†’ 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)⟢(𝒫 𝑋 ↑m 𝒫 𝑋))
20 dssmapclsntr.k . . . . 5 𝐾 = (clsβ€˜π½)
219, 20clselmap 43480 . . . 4 (𝐽 ∈ Top β†’ 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
2219, 21ffvelcdmd 7089 . . 3 (𝐽 ∈ Top β†’ (π·β€˜πΎ) ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
23 elmapfn 8875 . . 3 ((π·β€˜πΎ) ∈ (𝒫 𝑋 ↑m 𝒫 𝑋) β†’ (π·β€˜πΎ) Fn 𝒫 𝑋)
2422, 23syl 17 . 2 (𝐽 ∈ Top β†’ (π·β€˜πΎ) Fn 𝒫 𝑋)
25 elpwi 4605 . . . . 5 (𝑑 ∈ 𝒫 𝑋 β†’ 𝑑 βŠ† 𝑋)
269ntrval2 22942 . . . . 5 ((𝐽 ∈ Top ∧ 𝑑 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜π‘‘) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑑))))
2725, 26sylan2 592 . . . 4 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ ((intβ€˜π½)β€˜π‘‘) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑑))))
288fveq1i 6892 . . . 4 (πΌβ€˜π‘‘) = ((intβ€˜π½)β€˜π‘‘)
2920fveq1i 6892 . . . . 5 (πΎβ€˜(𝑋 βˆ– 𝑑)) = ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑑))
3029difeq2i 4115 . . . 4 (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– 𝑑))) = (𝑋 βˆ– ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝑑)))
3127, 28, 303eqtr4g 2792 . . 3 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ (πΌβ€˜π‘‘) = (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– 𝑑))))
3216adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ 𝑋 ∈ 𝐽)
3321adantr 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
34 eqid 2727 . . . 4 (π·β€˜πΎ) = (π·β€˜πΎ)
35 simpr 484 . . . 4 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ 𝑑 ∈ 𝒫 𝑋)
36 eqid 2727 . . . 4 ((π·β€˜πΎ)β€˜π‘‘) = ((π·β€˜πΎ)β€˜π‘‘)
3714, 15, 32, 33, 34, 35, 36dssmapfv3d 43372 . . 3 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ ((π·β€˜πΎ)β€˜π‘‘) = (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– 𝑑))))
3831, 37eqtr4d 2770 . 2 ((𝐽 ∈ Top ∧ 𝑑 ∈ 𝒫 𝑋) β†’ (πΌβ€˜π‘‘) = ((π·β€˜πΎ)β€˜π‘‘))
3913, 24, 38eqfnfvd 7037 1 (𝐽 ∈ Top β†’ 𝐼 = (π·β€˜πΎ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  Vcvv 3469   βˆ– cdif 3941   ∩ cin 3943   βŠ† wss 3944  π’« cpw 4598  βˆͺ cuni 4903   ↦ cmpt 5225   Fn wfn 6537  βŸΆwf 6538  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414   ↑m cmap 8836  Topctop 22782  intcnt 22908  clsccl 22909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8838  df-top 22783  df-cld 22910  df-ntr 22911  df-cls 22912
This theorem is referenced by:  dssmapclsntr  43482
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