Theorem ntrclsk4 44068

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 6866	. . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘(𝐼‘𝑠)) = (𝐼‘(𝐼‘𝑡)))
2		fveq2 6861	. . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
3	1, 2	eqeq12d 2746	. . 3 ⊢ (𝑠 = 𝑡 → ((𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ (𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡)))
4	3	cbvralvw 3216	. 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 44031	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
9	5, 6	ntrclsrcomplex 44031	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
11		difeq2 4086	. . . . . 6 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡)))
12	11	eqeq2d 2741	. . . . 5 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
13	12	adantl 481	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
14		elpwi 4573	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 ⊆ 𝐵)
15		dfss4 4235	. . . . . . 7 ⊢ (𝑡 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
16	14, 15	sylib 218	. . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
17	16	eqcomd 2736	. . . . 5 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡)))
18	17	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡)))
19	10, 13, 18	rspcedvd 3593	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ 𝑠))
20		2fveq3 6866	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘(𝐼‘𝑡)) = (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))))
21		fveq2 6861	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘𝑡) = (𝐼‘(𝐵 ∖ 𝑠)))
22	20, 21	eqeq12d 2746	. . . . 5 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → ((𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡) ↔ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠))))
23	22	3ad2ant3 1135	. . . 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 44049	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
26		elmapi 8825	. . . . . . . . . . 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	ffvelcdmd 7060	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
29	27, 28	ffvelcdmd 7060	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) ∈ 𝒫 𝐵)
30	29	elpwid 4575	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) ⊆ 𝐵)
31	28	elpwid 4575	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
32		rcompleq 4271	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝜑 → ((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
34	33	adantr 480	. . . . . 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 44048	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾𝐷𝐼)
36	35	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐾𝐷𝐼)
37	24, 5, 35	ntrclsiex 44049	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
38		elmapi 8825	. . . . . . . . . . 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	ffvelcdmda 7059	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘𝑠) ∈ 𝒫 𝐵)
41	24, 5, 36, 40	ntrclsfv 44055	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾‘𝑠)) = (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾‘𝑠)))))
42		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
43	24, 5, 36, 42	ntrclsfv 44055	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
44	43	difeq2d 4092	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾‘𝑠)) = (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
45		dfss4 4235	. . . . . . . . . . . . 13 ⊢ ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))) = (𝐼‘(𝐵 ∖ 𝑠)))
46	31, 45	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))) = (𝐼‘(𝐵 ∖ 𝑠)))
47	46	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))) = (𝐼‘(𝐵 ∖ 𝑠)))
48	44, 47	eqtrd 2765	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐾‘𝑠)) = (𝐼‘(𝐵 ∖ 𝑠)))
49	48	fveq2d 6865	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘(𝐵 ∖ (𝐾‘𝑠))) = (𝐼‘(𝐼‘(𝐵 ∖ 𝑠))))
50	49	difeq2d 4092	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐼‘(𝐵 ∖ (𝐾‘𝑠)))) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))))
51	41, 50	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐾‘(𝐾‘𝑠)) = (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))))
52	51, 43	eqeq12d 2746	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠) ↔ (𝐵 ∖ (𝐼‘(𝐼‘(𝐵 ∖ 𝑠)))) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
53	34, 52	bitr4d 282	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
54	53	3adant3 1132	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐼‘(𝐵 ∖ 𝑠))) = (𝐼‘(𝐵 ∖ 𝑠)) ↔ (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
55	23, 54	bitrd 279	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡) ↔ (𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
56	8, 19, 55	ralxfrd2 5370	. 2 ⊢ (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑡)) = (𝐼‘𝑡) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))
57	4, 56	bitrid 283	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝐾‘(𝐾‘𝑠)) = (𝐾‘𝑠)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ∖ cdif 3914   ⊆ wss 3917  𝒫 cpw 4566   class class class wbr 5110   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804

This theorem is referenced by: (None)