Theorem ntrclsk2 44649

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 6867	. . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
2		id 22	. . . 4 ⊢ (𝑠 = 𝑡 → 𝑠 = 𝑡)
3	1, 2	sseq12d 3970	. . 3 ⊢ (𝑠 = 𝑡 → ((𝐼‘𝑠) ⊆ 𝑠 ↔ (𝐼‘𝑡) ⊆ 𝑡))
4	3	cbvralvw 3241	. 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 44616	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
8	7	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
9	5, 6	ntrclsrcomplex 44616	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
10	9	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
11		difeq2 4075	. . . . . 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 485	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
14		elpwi 4563	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 ⊆ 𝐵)
15		dfss4 4222	. . . . . . 7 ⊢ (𝑡 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
16	14, 15	sylib 220	. . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
17	16	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
18	17	eqcomd 2769	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡)))
19	10, 13, 18	rspcedvd 3584	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ 𝑠))
20		fveq2 6867	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘𝑡) = (𝐼‘(𝐵 ∖ 𝑠)))
21		id 22	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → 𝑡 = (𝐵 ∖ 𝑠))
22	20, 21	sseq12d 3970	. . . . 5 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → ((𝐼‘𝑡) ⊆ 𝑡 ↔ (𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠)))
23	22	3ad2ant3 1149	. . . 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 44634	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
26		elmapi 8830	. . . . . . . . . . 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 1147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
29	7	3ad2ant1 1147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
30	28, 29	ffvelcdmd 7066	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
31	30	elpwid 4565	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
32		difssd 4091	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
33		sscon34b 4257	. . . . . . 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 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
35		simp2 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝑠 ∈ 𝒫 𝐵)
36		elpwi 4563	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝒫 𝐵 → 𝑠 ⊆ 𝐵)
37		dfss4 4222	. . . . . . . . 9 ⊢ (𝑠 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑠)) = 𝑠)
38	36, 37	sylib 220	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑠)) = 𝑠)
39	38	sseq1d 3968	. . . . . . 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 44615	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
43	42	3ad2ant1 1147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐵 ∈ V)
44	25	3ad2ant1 1147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
45		eqid 2763	. . . . . . 7 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
46		eqid 2763	. . . . . . 7 ⊢ ((𝐷‘𝐼)‘𝑠) = ((𝐷‘𝐼)‘𝑠)
47	24, 5, 43, 44, 45, 35, 46	dssmapfv3d 44600	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐷‘𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
48	47	sseq2d 3969	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
49	24, 5, 6	ntrclsfv1 44636	. . . . . . . 8 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
50	49	fveq1d 6869	. . . . . . 7 ⊢ (𝜑 → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
51	50	sseq2d 3969	. . . . . 6 ⊢ (𝜑 → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾‘𝑠)))
52	51	3ad2ant1 1147	. . . . 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 5370	. 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 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∀wral 3077  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  𝒫 cpw 4556   class class class wbr 5101   ↦ cmpt 5182  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ↑_m cmap 8808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810

This theorem is referenced by: (None)