Theorem dssmapntrcls 44090

Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 35175. (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 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))
dssmapclsntr.d	⊢ 𝐷 = (𝑂‘𝑋)

Assertion

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

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.

Step	Hyp	Ref	Expression
1		vpwex 5395	. . . . . . 7 ⊢ 𝒫 𝑡 ∈ V
2	1	inex2 5336	. . . . . 6 ⊢ (𝐽 ∩ 𝒫 𝑡) ∈ V
3	2	uniex 7776	. . . . 5 ⊢ ∪ (𝐽 ∩ 𝒫 𝑡) ∈ V
4	3	rgenw 3071	. . . 4 ⊢ ∀𝑡 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑡) ∈ V
5		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑡𝒫 𝑋
6	5	fnmptf 6716	. . . 4 ⊢ (∀𝑡 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑡) ∈ V → (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)) Fn 𝒫 𝑋)
7	4, 6	mp1i 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
10	9	ntrfval 23053	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)))
11	8, 10	eqtrid 2792	. . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)))
12	11	fneq1d 6672	. . 3 ⊢ (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)) Fn 𝒫 𝑋))
13	7, 12	mpbird 257	. 2 ⊢ (𝐽 ∈ Top → 𝐼 Fn 𝒫 𝑋)
14		dssmapclsntr.o	. . . . . 6 ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))
15		dssmapclsntr.d	. . . . . 6 ⊢ 𝐷 = (𝑂‘𝑋)
16	9	topopn 22933	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
17	14, 15, 16	dssmapf1od 43983	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑m 𝒫 𝑋))
18		f1of 6862	. . . . 5 ⊢ (𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑m 𝒫 𝑋) → 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)⟶(𝒫 𝑋 ↑m 𝒫 𝑋))
19	17, 18	syl 17	. . . 4 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)⟶(𝒫 𝑋 ↑m 𝒫 𝑋))
20		dssmapclsntr.k	. . . . 5 ⊢ 𝐾 = (cls‘𝐽)
21	9, 20	clselmap 44089	. . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
22	19, 21	ffvelcdmd 7119	. . 3 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
23		elmapfn 8923	. . 3 ⊢ ((𝐷‘𝐾) ∈ (𝒫 𝑋 ↑m 𝒫 𝑋) → (𝐷‘𝐾) Fn 𝒫 𝑋)
24	22, 23	syl 17	. 2 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) Fn 𝒫 𝑋)
25		elpwi 4629	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝑋 → 𝑡 ⊆ 𝑋)
26	9	ntrval2 23080	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ⊆ 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡))))
27	25, 26	sylan2 592	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡))))
28	8	fveq1i 6921	. . . 4 ⊢ (𝐼‘𝑡) = ((int‘𝐽)‘𝑡)
29	20	fveq1i 6921	. . . . 5 ⊢ (𝐾‘(𝑋 ∖ 𝑡)) = ((cls‘𝐽)‘(𝑋 ∖ 𝑡))
30	29	difeq2i 4146	. . . 4 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡)))
31	27, 28, 30	3eqtr4g 2805	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))))
32	16	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑋 ∈ 𝐽)
33	21	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
34		eqid 2740	. . . 4 ⊢ (𝐷‘𝐾) = (𝐷‘𝐾)
35		simpr 484	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑡 ∈ 𝒫 𝑋)
36		eqid 2740	. . . 4 ⊢ ((𝐷‘𝐾)‘𝑡) = ((𝐷‘𝐾)‘𝑡)
37	14, 15, 32, 33, 34, 35, 36	dssmapfv3d 43981	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((𝐷‘𝐾)‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))))
38	31, 37	eqtr4d 2783	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼‘𝑡) = ((𝐷‘𝐾)‘𝑡))
39	13, 24, 38	eqfnfvd 7067	1 ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931   ↦ cmpt 5249   Fn wfn 6568  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  Topctop 22920  intcnt 23046  clsccl 23047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886  df-top 22921  df-cld 23048  df-ntr 23049  df-cls 23050

This theorem is referenced by:  dssmapclsntr  44091