Theorem dssmapntrcls 39879

Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 32076. (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 5127	. . . . . . 7 ⊢ 𝒫 𝑡 ∈ V
2	1	inex2 5075	. . . . . 6 ⊢ (𝐽 ∩ 𝒫 𝑡) ∈ V
3	2	uniex 7281	. . . . 5 ⊢ ∪ (𝐽 ∩ 𝒫 𝑡) ∈ V
4	3	rgenw 3093	. . . 4 ⊢ ∀𝑡 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑡) ∈ V
5		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑡𝒫 𝑋
6	5	fnmptf 6311	. . . 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 21351	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)))
11	8, 10	syl5eq 2819	. . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)))
12	11	fneq1d 6276	. . 3 ⊢ (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)) Fn 𝒫 𝑋))
13	7, 12	mpbird 249	. 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 21233	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
17	14, 15, 16	dssmapf1od 39768	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑𝑚 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑𝑚 𝒫 𝑋))
18		f1of 6441	. . . . 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 39878	. . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑𝑚 𝒫 𝑋))
22	19, 21	ffvelrnd 6675	. . 3 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) ∈ (𝒫 𝑋 ↑𝑚 𝒫 𝑋))
23		elmapfn 8227	. . 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 4426	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝑋 → 𝑡 ⊆ 𝑋)
26	9	ntrval2 21378	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ⊆ 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡))))
27	25, 26	sylan2 584	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡))))
28	8	fveq1i 6497	. . . 4 ⊢ (𝐼‘𝑡) = ((int‘𝐽)‘𝑡)
29	20	fveq1i 6497	. . . . 5 ⊢ (𝐾‘(𝑋 ∖ 𝑡)) = ((cls‘𝐽)‘(𝑋 ∖ 𝑡))
30	29	difeq2i 3979	. . . 4 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡)))
31	27, 28, 30	3eqtr4g 2832	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))))
32	16	adantr 473	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑋 ∈ 𝐽)
33	21	adantr 473	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝐾 ∈ (𝒫 𝑋 ↑𝑚 𝒫 𝑋))
34		eqid 2771	. . . 4 ⊢ (𝐷‘𝐾) = (𝐷‘𝐾)
35		simpr 477	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑡 ∈ 𝒫 𝑋)
36		eqid 2771	. . . 4 ⊢ ((𝐷‘𝐾)‘𝑡) = ((𝐷‘𝐾)‘𝑡)
37	14, 15, 32, 33, 34, 35, 36	dssmapfv3d 39766	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((𝐷‘𝐾)‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))))
38	31, 37	eqtr4d 2810	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼‘𝑡) = ((𝐷‘𝐾)‘𝑡))
39	13, 24, 38	eqfnfvd 6628	1 ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051  ∀wral 3081  Vcvv 3408   ∖ cdif 3819   ∩ cin 3821   ⊆ wss 3822  𝒫 cpw 4416  ∪ cuni 4708   ↦ cmpt 5004   Fn wfn 6180  ⟶wf 6181  –1-1-onto→wf1o 6184  ‘cfv 6185  (class class class)co 6974   ↑_𝑚 cmap 8204  Topctop 21220  intcnt 21344  clsccl 21345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-iin 4791  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7499  df-2nd 7500  df-map 8206  df-top 21221  df-cld 21346  df-ntr 21347  df-cls 21348

This theorem is referenced by:  dssmapclsntr  39880