Theorem ntrclsk2 44582

Description: An interior function is contracting if and only if the closure function is expansive. (Contributed by RP, 9-Jun-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
ntrclsk2	⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))

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

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

Proof of Theorem ntrclsk2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6852	. . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
2		id 22	. . . 4 ⊢ (𝑠 = 𝑡 → 𝑠 = 𝑡)
3	1, 2	sseq12d 3960	. . 3 ⊢ (𝑠 = 𝑡 → ((𝐼‘𝑠) ⊆ 𝑠 ↔ (𝐼‘𝑡) ⊆ 𝑡))
4	3	cbvralvw 3230	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑡 ∈ 𝒫 𝐵(𝐼‘𝑡) ⊆ 𝑡)
5		ntrcls.d	. . . . 5 ⊢ 𝐷 = (𝑂‘𝐵)
6		ntrcls.r	. . . . 5 ⊢ (𝜑 → 𝐼𝐷𝐾)
7	5, 6	ntrclsrcomplex 44549	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
8	7	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
9	5, 6	ntrclsrcomplex 44549	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
10	9	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
11		difeq2 4065	. . . . . 6 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡)))
12	11	eqeq2d 2763	. . . . 5 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
13	12	adantl 484	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
14		elpwi 4552	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 ⊆ 𝐵)
15		dfss4 4212	. . . . . . 7 ⊢ (𝑡 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
16	14, 15	sylib 220	. . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
17	16	adantl 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
18	17	eqcomd 2758	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡)))
19	10, 13, 18	rspcedvd 3574	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ 𝑠))
20		fveq2 6852	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘𝑡) = (𝐼‘(𝐵 ∖ 𝑠)))
21		id 22	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → 𝑡 = (𝐵 ∖ 𝑠))
22	20, 21	sseq12d 3960	. . . . 5 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → ((𝐼‘𝑡) ⊆ 𝑡 ↔ (𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠)))
23	22	3ad2ant3 1144	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘𝑡) ⊆ 𝑡 ↔ (𝐼‘(𝐵 ∖ 𝑠)) ⊆ (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 44567	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
26		elmapi 8815	. . . . . . . . . . 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	3ad2ant1 1142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
29	7	3ad2ant1 1142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
30	28, 29	ffvelcdmd 7051	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
31	30	elpwid 4554	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
32		difssd 4081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
33		sscon34b 4247	. . . . . . 7 ⊢ (((𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵 ∧ (𝐵 ∖ 𝑠) ⊆ 𝐵) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
34	31, 32, 33	syl2anc 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
35		simp2 1146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝑠 ∈ 𝒫 𝐵)
36		elpwi 4552	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝒫 𝐵 → 𝑠 ⊆ 𝐵)
37		dfss4 4212	. . . . . . . . 9 ⊢ (𝑠 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑠)) = 𝑠)
38	36, 37	sylib 220	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑠)) = 𝑠)
39	38	sseq1d 3958	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝐵 → ((𝐵 ∖ (𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
40	35, 39	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐵 ∖ (𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
41	34, 40	bitrd 281	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
42	5, 6	ntrclsbex 44548	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
43	42	3ad2ant1 1142	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐵 ∈ V)
44	25	3ad2ant1 1142	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
45		eqid 2752	. . . . . . 7 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
46		eqid 2752	. . . . . . 7 ⊢ ((𝐷‘𝐼)‘𝑠) = ((𝐷‘𝐼)‘𝑠)
47	24, 5, 43, 44, 45, 35, 46	dssmapfv3d 44533	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐷‘𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
48	47	sseq2d 3959	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
49	24, 5, 6	ntrclsfv1 44569	. . . . . . . 8 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
50	49	fveq1d 6854	. . . . . . 7 ⊢ (𝜑 → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
51	50	sseq2d 3959	. . . . . 6 ⊢ (𝜑 → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾‘𝑠)))
52	51	3ad2ant1 1142	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾‘𝑠)))
53	41, 48, 52	3bitr2d 309	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ 𝑠 ⊆ (𝐾‘𝑠)))
54	23, 53	bitrd 281	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘𝑡) ⊆ 𝑡 ↔ 𝑠 ⊆ (𝐾‘𝑠)))
55	8, 19, 54	ralxfrd2 5359	. 2 ⊢ (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘𝑡) ⊆ 𝑡 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))
56	4, 55	bitrid 285	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132  ∀wral 3066  Vcvv 3444   ∖ cdif 3892   ⊆ wss 3895  𝒫 cpw 4545   class class class wbr 5090   ↦ cmpt 5171  ⟶wf 6502  ‘cfv 6506  (class class class)co 7381   ↑_m cmap 8792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-1st 7955  df-2nd 7956  df-map 8794

This theorem is referenced by: (None)