Theorem dssmapntrcls 44405

Description: The interior and closure operators on a topology are duals of each other. See also kur14lem2 35382. (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 5323	. . . . . . 7 ⊢ 𝒫 𝑡 ∈ V
2	1	inex2 5264	. . . . . 6 ⊢ (𝐽 ∩ 𝒫 𝑡) ∈ V
3	2	uniex 7688	. . . . 5 ⊢ ∪ (𝐽 ∩ 𝒫 𝑡) ∈ V
4	3	rgenw 3056	. . . 4 ⊢ ∀𝑡 ∈ 𝒫 𝑋∪ (𝐽 ∩ 𝒫 𝑡) ∈ V
5		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑡𝒫 𝑋
6	5	fnmptf 6629	. . . 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 22972	. . . . 5 ⊢ (𝐽 ∈ Top → (int‘𝐽) = (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)))
11	8, 10	eqtrid 2784	. . . 4 ⊢ (𝐽 ∈ Top → 𝐼 = (𝑡 ∈ 𝒫 𝑋 ↦ ∪ (𝐽 ∩ 𝒫 𝑡)))
12	11	fneq1d 6586	. . 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 22854	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
17	14, 15, 16	dssmapf1od 44298	. . . . 5 ⊢ (𝐽 ∈ Top → 𝐷:(𝒫 𝑋 ↑m 𝒫 𝑋)–1-1-onto→(𝒫 𝑋 ↑m 𝒫 𝑋))
18		f1of 6775	. . . . 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 44404	. . . 4 ⊢ (𝐽 ∈ Top → 𝐾 ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
22	19, 21	ffvelcdmd 7032	. . 3 ⊢ (𝐽 ∈ Top → (𝐷‘𝐾) ∈ (𝒫 𝑋 ↑m 𝒫 𝑋))
23		elmapfn 8806	. . 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 4562	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝑋 → 𝑡 ⊆ 𝑋)
26	9	ntrval2 22999	. . . . 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 6836	. . . 4 ⊢ (𝐼‘𝑡) = ((int‘𝐽)‘𝑡)
29	20	fveq1i 6836	. . . . 5 ⊢ (𝐾‘(𝑋 ∖ 𝑡)) = ((cls‘𝐽)‘(𝑋 ∖ 𝑡))
30	29	difeq2i 4076	. . . 4 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑡)))
31	27, 28, 30	3eqtr4g 2797	. . 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 2737	. . . 4 ⊢ (𝐷‘𝐾) = (𝐷‘𝐾)
35		simpr 484	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → 𝑡 ∈ 𝒫 𝑋)
36		eqid 2737	. . . 4 ⊢ ((𝐷‘𝐾)‘𝑡) = ((𝐷‘𝐾)‘𝑡)
37	14, 15, 32, 33, 34, 35, 36	dssmapfv3d 44296	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → ((𝐷‘𝐾)‘𝑡) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝑡))))
38	31, 37	eqtr4d 2775	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑡 ∈ 𝒫 𝑋) → (𝐼‘𝑡) = ((𝐷‘𝐾)‘𝑡))
39	13, 24, 38	eqfnfvd 6981	1 ⊢ (𝐽 ∈ Top → 𝐼 = (𝐷‘𝐾))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3441   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4555  ∪ cuni 4864   ↦ cmpt 5180   Fn wfn 6488  ⟶wf 6489  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360   ↑_m cmap 8767  Topctop 22841  intcnt 22965  clsccl 22966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8769  df-top 22842  df-cld 22967  df-ntr 22968  df-cls 22969

This theorem is referenced by:  dssmapclsntr  44406