Theorem dssmapntrcls 43181

Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 34496. (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 5374	. . . . . . 7 ⊢ 𝒫 𝑡 ∈ V
2	1	inex2 5317	. . . . . 6 ⊢ (𝐽 ∩ 𝒫 𝑡) ∈ V
3	2	uniex 7733	. . . . 5 ⊢ ∪ (𝐽 ∩ 𝒫 𝑡) ∈ V
4	3	rgenw 3063	. . . 4 ⊢ ∀𝑡 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑡) ∈ V
5		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑡𝒫 𝑋
6	5	fnmptf 6685	. . . 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 22748	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)))
11	8, 10	eqtrid 2782	. . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)))
12	11	fneq1d 6641	. . 3 ⊢ (𝐽 ∈ Top → (𝐼 Fn 𝒫 𝑋 ↔ (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)) Fn 𝒫 𝑋))
13	7, 12	mpbird 256	. 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 22628	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
17	14, 15, 16	dssmapf1od 43074	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑m 𝒫 𝑋))
18		f1of 6832	. . . . 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 43180	. . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
22	19, 21	ffvelcdmd 7086	. . 3 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
23		elmapfn 8861	. . 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 4608	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝑋 → 𝑡 ⊆ 𝑋)
26	9	ntrval2 22775	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ⊆ 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡))))
27	25, 26	sylan2 591	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((int‘𝐽)‘𝑡) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡))))
28	8	fveq1i 6891	. . . 4 ⊢ (𝐼‘𝑡) = ((int‘𝐽)‘𝑡)
29	20	fveq1i 6891	. . . . 5 ⊢ (𝐾‘(𝑋 ∖ 𝑡)) = ((cls‘𝐽)‘(𝑋 ∖ 𝑡))
30	29	difeq2i 4118	. . . 4 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡)))
31	27, 28, 30	3eqtr4g 2795	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))))
32	16	adantr 479	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑋 ∈ 𝐽)
33	21	adantr 479	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
34		eqid 2730	. . . 4 ⊢ (𝐷‘𝐾) = (𝐷‘𝐾)
35		simpr 483	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑡 ∈ 𝒫 𝑋)
36		eqid 2730	. . . 4 ⊢ ((𝐷‘𝐾)‘𝑡) = ((𝐷‘𝐾)‘𝑡)
37	14, 15, 32, 33, 34, 35, 36	dssmapfv3d 43072	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((𝐷‘𝐾)‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))))
38	31, 37	eqtr4d 2773	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼‘𝑡) = ((𝐷‘𝐾)‘𝑡))
39	13, 24, 38	eqfnfvd 7034	1 ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059  Vcvv 3472   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4601  ∪ cuni 4907   ↦ cmpt 5230   Fn wfn 6537  ⟶wf 6538  –1-1-onto→wf1o 6541  ‘cfv 6542  (class class class)co 7411   ↑_m cmap 8822  Topctop 22615  intcnt 22741  clsccl 22742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-top 22616  df-cld 22743  df-ntr 22744  df-cls 22745

This theorem is referenced by:  dssmapclsntr  43182