Theorem ntrclsk4 38370

Description: Idempotence of the interior function is equivalent to idempotence of the closure function. (Contributed by RP, 10-Jul-2021.)

Hypotheses

Ref	Expression
ntrcls.o	⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
ntrcls.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrcls.r	⊢ (𝜑 → 𝐼𝐷𝐾)

Assertion

Ref	Expression
ntrclsk4	⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑠   𝑗,𝐼,𝑘,𝑠   𝑗,𝐾   𝜑,𝑖,𝑗,𝑘,𝑠

Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘,𝑠)   𝐼(𝑖)   𝐾(𝑖,𝑘,𝑠)   𝑂(𝑖,𝑗,𝑘,𝑠)

Proof of Theorem ntrclsk4

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
2	1	fveq2d 6195	. . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘(𝐼‘𝑠)) = (𝐼‘(𝐼‘𝑡)))
3	2, 1	eqeq12d 2637	. . 3 ⊢ (𝑠 = 𝑡 → ((𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ (𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡)))
4	3	cbvralv 3171	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡))
5		ntrcls.d	. . . . 5 ⊢ 𝐷 = (𝑂‘𝐵)
6		ntrcls.r	. . . . 5 ⊢ (𝜑 → 𝐼𝐷𝐾)
7	5, 6	ntrclsrcomplex 38333	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
8	7	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
9	5, 6	ntrclsrcomplex 38333	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
10	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
11		difeq2 3722	. . . . . 6 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡)))
12	11	eqeq2d 2632	. . . . 5 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
13	12	adantl 482	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
14		elpwi 4168	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 ⊆ 𝐵)
15		dfss4 3858	. . . . . . 7 ⊢ (𝑡 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
16	14, 15	sylib 208	. . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
17	16	eqcomd 2628	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡)))
18	17	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡)))
19	10, 13, 18	rspcedvd 3317	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ 𝑠))
20		fveq2 6191	. . . . . . 7 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘𝑡) = (𝐼‘(𝐵 ∖ 𝑠)))
21	20	fveq2d 6195	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘(𝐼‘𝑡)) = (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))))
22	21, 20	eqeq12d 2637	. . . . 5 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → ((𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡) ↔ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠))))
23	22	3ad2ant3 1084	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡) ↔ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠))))
24		ntrcls.o	. . . . . . . . . . . 12 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
25	24, 5, 6	ntrclsiex 38351	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
26		elmapi 7879	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
27	25, 26	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
28	27, 7	ffvelrnd 6360	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
29	27, 28	ffvelrnd 6360	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) ∈ 𝒫 𝐵)
30	29	elpwid 4170	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) ⊆ 𝐵)
31	28	elpwid 4170	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
32		rcompleq 38318	. . . . . . . 8 ⊢ (((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) ⊆ 𝐵 ∧ (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵) → ((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
33	30, 31, 32	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → ((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
34	33	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
35	24, 5, 6	ntrclsnvobr 38350	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾𝐷𝐼)
36	35	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐾𝐷𝐼)
37	24, 5, 35	ntrclsiex 38351	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
38		elmapi 7879	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
40	39	ffvelrnda 6359	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘𝑠) ∈ 𝒫 𝐵)
41	24, 5, 36, 40	ntrclsfv 38357	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾‘𝑠)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾‘𝑠)))))
42		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
43	24, 5, 36, 42	ntrclsfv 38357	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
44	43	difeq2d 3728	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾‘𝑠)) = (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
45		dfss4 3858	. . . . . . . . . . . . 13 ⊢ ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))) = (𝐼‘(𝐵 ∖ 𝑠)))
46	31, 45	sylib 208	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))) = (𝐼‘(𝐵 ∖ 𝑠)))
47	46	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))) = (𝐼‘(𝐵 ∖ 𝑠)))
48	44, 47	eqtrd 2656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾‘𝑠)) = (𝐼‘(𝐵 ∖ 𝑠)))
49	48	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘(𝐵 ∖ (𝐾‘𝑠))) = (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))))
50	49	difeq2d 3728	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾‘𝑠)))) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))))
51	41, 50	eqtrd 2656	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾‘𝑠)) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))))
52	51, 43	eqeq12d 2637	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
53	34, 52	bitr4d 271	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
54	53	3adant3 1081	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
55	23, 54	bitrd 268	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡) ↔ (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
56	8, 19, 55	ralxfrd2 4884	. 2 ⊢ (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
57	4, 56	syl5bb 272	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  𝒫 cpw 4158   class class class wbr 4653   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by: (None)