Theorem ntrclsk2 44030

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 6920	. . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
2		id 22	. . . 4 ⊢ (𝑠 = 𝑡 → 𝑠 = 𝑡)
3	1, 2	sseq12d 4042	. . 3 ⊢ (𝑠 = 𝑡 → ((𝐼‘𝑠) ⊆ 𝑠 ↔ (𝐼‘𝑡) ⊆ 𝑡))
4	3	cbvralvw 3243	. 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 43997	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
9	5, 6	ntrclsrcomplex 43997	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
11		difeq2 4143	. . . . . 6 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡)))
12	11	eqeq2d 2751	. . . . 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 4629	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 ⊆ 𝐵)
15		dfss4 4288	. . . . . . 7 ⊢ (𝑡 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
16	14, 15	sylib 218	. . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
17	16	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
18	17	eqcomd 2746	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡)))
19	10, 13, 18	rspcedvd 3637	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ 𝑠))
20		fveq2 6920	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘𝑡) = (𝐼‘(𝐵 ∖ 𝑠)))
21		id 22	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → 𝑡 = (𝐵 ∖ 𝑠))
22	20, 21	sseq12d 4042	. . . . 5 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → ((𝐼‘𝑡) ⊆ 𝑡 ↔ (𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠)))
23	22	3ad2ant3 1135	. . . 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 44015	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
26		elmapi 8907	. . . . . . . . . . 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 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
29	7	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
30	28, 29	ffvelcdmd 7119	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
31	30	elpwid 4631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
32		difssd 4160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
33		sscon34b 4323	. . . . . . 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 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
35		simp2 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝑠 ∈ 𝒫 𝐵)
36		elpwi 4629	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝒫 𝐵 → 𝑠 ⊆ 𝐵)
37		dfss4 4288	. . . . . . . . 9 ⊢ (𝑠 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑠)) = 𝑠)
38	36, 37	sylib 218	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑠)) = 𝑠)
39	38	sseq1d 4040	. . . . . . 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 279	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
42	5, 6	ntrclsbex 43996	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
43	42	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐵 ∈ V)
44	25	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
45		eqid 2740	. . . . . . 7 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
46		eqid 2740	. . . . . . 7 ⊢ ((𝐷‘𝐼)‘𝑠) = ((𝐷‘𝐼)‘𝑠)
47	24, 5, 43, 44, 45, 35, 46	dssmapfv3d 43981	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐷‘𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
48	47	sseq2d 4041	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
49	24, 5, 6	ntrclsfv1 44017	. . . . . . . 8 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
50	49	fveq1d 6922	. . . . . . 7 ⊢ (𝜑 → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
51	50	sseq2d 4041	. . . . . 6 ⊢ (𝜑 → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾‘𝑠)))
52	51	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾‘𝑠)))
53	41, 48, 52	3bitr2d 307	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ 𝑠 ⊆ (𝐾‘𝑠)))
54	23, 53	bitrd 279	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘𝑡) ⊆ 𝑡 ↔ 𝑠 ⊆ (𝐾‘𝑠)))
55	8, 19, 54	ralxfrd2 5430	. 2 ⊢ (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘𝑡) ⊆ 𝑡 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))
56	4, 55	bitrid 283	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  𝒫 cpw 4622   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886

This theorem is referenced by: (None)