ntrclsk2 - Mathbox for Richard Penner

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Theorem ntrclsk2 42495

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 6862	. . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
2		id 22	. . . 4 ⊢ (𝑠 = 𝑡 → 𝑠 = 𝑡)
3	1, 2	sseq12d 3995	. . 3 ⊢ (𝑠 = 𝑡 → ((𝐼‘𝑠) ⊆ 𝑠 ↔ (𝐼‘𝑡) ⊆ 𝑡))
4	3	cbvralvw 3233	. 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 42462	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
8	7	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
9	5, 6	ntrclsrcomplex 42462	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
10	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
11		difeq2 4096	. . . . . 6 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡)))
12	11	eqeq2d 2742	. . . . 5 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
13	12	adantl 482	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
14		elpwi 4587	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 ⊆ 𝐵)
15		dfss4 4238	. . . . . . 7 ⊢ (𝑡 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
16	14, 15	sylib 217	. . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
17	16	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
18	17	eqcomd 2737	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡)))
19	10, 13, 18	rspcedvd 3597	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ 𝑠))
20		fveq2 6862	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘𝑡) = (𝐼‘(𝐵 ∖ 𝑠)))
21		id 22	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → 𝑡 = (𝐵 ∖ 𝑠))
22	20, 21	sseq12d 3995	. . . . 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 42480	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑_m 𝒫 𝐵))
26		elmapi 8809	. . . . . . . . . . 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 7056	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐼‘(𝐵 ∖ 𝑠)) ∈ 𝒫 𝐵)
31	30	elpwid 4589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐼‘(𝐵 ∖ 𝑠)) ⊆ 𝐵)
32		difssd 4112	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
33		sscon34b 4274	. . . . . . 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 584	. . . . . 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 4587	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝒫 𝐵 → 𝑠 ⊆ 𝐵)
37		dfss4 4238	. . . . . . . . 9 ⊢ (𝑠 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑠)) = 𝑠)
38	36, 37	sylib 217	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑠)) = 𝑠)
39	38	sseq1d 3993	. . . . . . 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 278	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
42	5, 6	ntrclsbex 42461	. . . . . . . 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 2731	. . . . . . 7 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
46		eqid 2731	. . . . . . 7 ⊢ ((𝐷‘𝐼)‘𝑠) = ((𝐷‘𝐼)‘𝑠)
47	24, 5, 43, 44, 45, 35, 46	dssmapfv3d 42446	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐷‘𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
48	47	sseq2d 3994	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
49	24, 5, 6	ntrclsfv1 42482	. . . . . . . 8 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
50	49	fveq1d 6864	. . . . . . 7 ⊢ (𝜑 → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
51	50	sseq2d 3994	. . . . . 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 306	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ 𝑠 ⊆ (𝐾‘𝑠)))
54	23, 53	bitrd 278	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘𝑡) ⊆ 𝑡 ↔ 𝑠 ⊆ (𝐾‘𝑠)))
55	8, 19, 54	ralxfrd2 5387	. 2 ⊢ (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘𝑡) ⊆ 𝑡 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))
56	4, 55	bitrid 282	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3060 Vcvv 3459 ∖ cdif 3925 ⊆ wss 3928 𝒫 cpw 4580 class class class wbr 5125 ↦ cmpt 5208 ⟶wf 6512 ‘cfv 6516 (class class class)co 7377 ↑_m cmap 8787

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-ov 7380 df-oprab 7381 df-mpo 7382 df-1st 7941 df-2nd 7942 df-map 8789

This theorem is referenced by: (None)

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