Theorem ntrclsk4 45264

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		2fveq3 6913	. . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘(𝐼‘𝑠)) = (𝐼‘(𝐼‘𝑡)))
2		fveq2 6908	. . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
3	1, 2	eqeq12d 2779	. . 3 ⊢ (𝑠 = 𝑡 → ((𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ (𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡)))
4	3	cbvralvw 3244	. 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 45227	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
8	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
9	5, 6	ntrclsrcomplex 45227	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
10	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
11		difeq2 4060	. . . . . 6 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡)))
12	11	eqeq2d 2774	. . . . 5 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
13	12	adantl 487	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
14		elpwi 4567	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 ⊆ 𝐵)
15		dfss4 4217	. . . . . . 7 ⊢ (𝑡 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
16	14, 15	sylib 221	. . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
17	16	eqcomd 2769	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡)))
18	17	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡)))
19	10, 13, 18	rspcedvd 3582	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ 𝑠))
20		2fveq3 6913	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘(𝐼‘𝑡)) = (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))))
21		fveq2 6908	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘𝑡) = (𝐼‘(𝐵 ∖ 𝑠)))
22	20, 21	eqeq12d 2779	. . . . 5 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → ((𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡) ↔ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠))))
23	22	3ad2ant3 1153	. . . 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 45245	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
26		elmapi 8897	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
27	25, 26	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
28	27, 7	ffvelcdmd 7109	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
29	27, 28	ffvelcdmd 7109	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) ∈ 𝒫 𝐵)
30	29	elpwid 4569	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) ⊆ 𝐵)
31	28	elpwid 4569	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
32		rcompleq 4253	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝜑 → ((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
34	33	adantr 486	. . . . . 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 45244	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾𝐷𝐼)
36	35	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐾𝐷𝐼)
37	24, 5, 35	ntrclsiex 45245	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
38		elmapi 8897	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
39	37, 38	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
40	39	ffvelcdmda 7108	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘𝑠) ∈ 𝒫 𝐵)
41	24, 5, 36, 40	ntrclsfv 45251	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾‘𝑠)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾‘𝑠)))))
42		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
43	24, 5, 36, 42	ntrclsfv 45251	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
44	43	difeq2d 4066	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾‘𝑠)) = (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
45		dfss4 4217	. . . . . . . . . . . . 13 ⊢ ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))) = (𝐼‘(𝐵 ∖ 𝑠)))
46	31, 45	sylib 221	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))) = (𝐼‘(𝐵 ∖ 𝑠)))
47	46	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))) = (𝐼‘(𝐵 ∖ 𝑠)))
48	44, 47	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾‘𝑠)) = (𝐼‘(𝐵 ∖ 𝑠)))
49	48	fveq2d 6912	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘(𝐵 ∖ (𝐾‘𝑠))) = (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))))
50	49	difeq2d 4066	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾‘𝑠)))) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))))
51	41, 50	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾‘𝑠)) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))))
52	51, 43	eqeq12d 2779	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
53	34, 52	bitr4d 285	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
54	53	3adant3 1150	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
55	23, 54	bitrd 282	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡) ↔ (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
56	8, 19, 55	ralxfrd2 5378	. 2 ⊢ (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
57	4, 56	bitrid 286	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  𝒫 cpw 4560   class class class wbr 5106   ↦ cmpt 5189  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ↑_m cmap 8875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-map 8877

This theorem is used by: (None)