Theorem dssmapntrcls 42879

Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 34198. (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 5376	. . . . . . 7 ⊢ 𝒫 𝑡 ∈ V
2	1	inex2 5319	. . . . . 6 ⊢ (𝐽 ∩ 𝒫 𝑡) ∈ V
3	2	uniex 7731	. . . . 5 ⊢ ∪ (𝐽 ∩ 𝒫 𝑡) ∈ V
4	3	rgenw 3066	. . . 4 ⊢ ∀𝑡 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑡) ∈ V
5		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑡𝒫 𝑋
6	5	fnmptf 6687	. . . 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 22528	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)))
11	8, 10	eqtrid 2785	. . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)))
12	11	fneq1d 6643	. . 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 22408	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
17	14, 15, 16	dssmapf1od 42772	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑m 𝒫 𝑋))
18		f1of 6834	. . . . 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 42878	. . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
22	19, 21	ffvelcdmd 7088	. . 3 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
23		elmapfn 8859	. . 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 4610	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝑋 → 𝑡 ⊆ 𝑋)
26	9	ntrval2 22555	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ⊆ 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡))))
27	25, 26	sylan2 594	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡))))
28	8	fveq1i 6893	. . . 4 ⊢ (𝐼‘𝑡) = ((int‘𝐽)‘𝑡)
29	20	fveq1i 6893	. . . . 5 ⊢ (𝐾‘(𝑋 ∖ 𝑡)) = ((cls‘𝐽)‘(𝑋 ∖ 𝑡))
30	29	difeq2i 4120	. . . 4 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡)))
31	27, 28, 30	3eqtr4g 2798	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))))
32	16	adantr 482	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑋 ∈ 𝐽)
33	21	adantr 482	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
34		eqid 2733	. . . 4 ⊢ (𝐷‘𝐾) = (𝐷‘𝐾)
35		simpr 486	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑡 ∈ 𝒫 𝑋)
36		eqid 2733	. . . 4 ⊢ ((𝐷‘𝐾)‘𝑡) = ((𝐷‘𝐾)‘𝑡)
37	14, 15, 32, 33, 34, 35, 36	dssmapfv3d 42770	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((𝐷‘𝐾)‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))))
38	31, 37	eqtr4d 2776	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼‘𝑡) = ((𝐷‘𝐾)‘𝑡))
39	13, 24, 38	eqfnfvd 7036	1 ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ∖ cdif 3946   ∩ cin 3948   ⊆ wss 3949  𝒫 cpw 4603  ∪ cuni 4909   ↦ cmpt 5232   Fn wfn 6539  ⟶wf 6540  –1-1-onto→wf1o 6543  ‘cfv 6544  (class class class)co 7409   ↑_m cmap 8820  Topctop 22395  intcnt 22521  clsccl 22522

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-top 22396  df-cld 22523  df-ntr 22524  df-cls 22525

This theorem is referenced by:  dssmapclsntr  42880