Theorem dssmapclsntr 41739

Description: The closure and interior operators on a topology are duals of each other. See also kur14lem2 33169. (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 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))
dssmapclsntr.d	⊢ 𝐷 = (𝑂‘𝑋)

Assertion

Ref	Expression
dssmapclsntr	⊢ (𝐽 ∈ Top → 𝐾 = (𝐷‘𝐼))

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

Step	Hyp	Ref	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 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))
5		dssmapclsntr.d	. . . . 5 ⊢ 𝐷 = (𝑂‘𝑋)
6	1, 2, 3, 4, 5	dssmapntrcls 41738	. . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾))
7	6	eqcomd 2744	. . 3 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) = 𝐼)
8	1	topopn 22055	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
9	4, 5, 8	dssmapf1od 41629	. . . 4 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑m 𝒫 𝑋))
10	1, 2	clselmap 41737	. . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
11		f1ocnvfv 7150	. . . 4 ⊢ ((𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑m 𝒫 𝑋) ∧ 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋)) → ((𝐷‘𝐾) = 𝐼 → (◡𝐷‘𝐼) = 𝐾))
12	9, 10, 11	syl2anc 584	. . 3 ⊢ (𝐽 ∈ Top → ((𝐷‘𝐾) = 𝐼 → (◡𝐷‘𝐼) = 𝐾))
13	7, 12	mpd 15	. 2 ⊢ (𝐽 ∈ Top → (◡𝐷‘𝐼) = 𝐾)
14	4, 5, 8	dssmapnvod 41628	. . 3 ⊢ (𝐽 ∈ Top → ◡𝐷 = 𝐷)
15	14	fveq1d 6776	. 2 ⊢ (𝐽 ∈ Top → (◡𝐷‘𝐼) = (𝐷‘𝐼))
16	13, 15	eqtr3d 2780	1 ⊢ (𝐽 ∈ Top → 𝐾 = (𝐷‘𝐼))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884  𝒫 cpw 4533  ∪ cuni 4839   ↦ cmpt 5157  ^◡ccnv 5588  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  Topctop 22042  intcnt 22168  clsccl 22169

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-top 22043  df-cld 22170  df-ntr 22171  df-cls 22172

This theorem is referenced by: (None)