Theorem ntrclsk2 45012

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 6873	. . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
2		id 23	. . . 4 ⊢ (𝑠 = 𝑡 → 𝑠 = 𝑡)
3	1, 2	sseq12d 3963	. . 3 ⊢ (𝑠 = 𝑡 → ((𝐼‘𝑠) ⊆ 𝑠 ↔ (𝐼‘𝑡) ⊆ 𝑡))
4	3	cbvralvw 3240	. 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 44979	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
8	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
9	5, 6	ntrclsrcomplex 44979	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
10	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
11		difeq2 4067	. . . . . 6 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡)))
12	11	eqeq2d 2771	. . . . 5 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝑡 = (𝐵 ∖ 𝑠) ↔ 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡))))
13	12	adantl 487	. . . 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 4214	. . . . . . 7 ⊢ (𝑡 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
16	14, 15	sylib 221	. . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
17	16	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
18	17	eqcomd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → 𝑡 = (𝐵 ∖ (𝐵 ∖ 𝑡)))
19	10, 13, 18	rspcedvd 3578	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ 𝑠))
20		fveq2 6873	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘𝑡) = (𝐼‘(𝐵 ∖ 𝑠)))
21		id 23	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → 𝑡 = (𝐵 ∖ 𝑠))
22	20, 21	sseq12d 3963	. . . . 5 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → ((𝐼‘𝑡) ⊆ 𝑡 ↔ (𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠)))
23	22	3ad2ant3 1153	. . . 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 44997	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
26		elmapi 8847	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
27	25, 26	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
28	27	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
29	7	3ad2ant1 1151	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
30	28, 29	ffvelcdmd 7073	. . . . . . . 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 4083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝐵 ∖ 𝑠) ⊆ 𝐵)
33		sscon34b 4249	. . . . . . 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 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ (𝐵 ∖ (𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
35		simp2 1155	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝑠 ∈ 𝒫 𝐵)
36		elpwi 4563	. . . . . . . . 9 ⊢ (𝑠 ∈ 𝒫 𝐵 → 𝑠 ⊆ 𝐵)
37		dfss4 4214	. . . . . . . . 9 ⊢ (𝑠 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑠)) = 𝑠)
38	36, 37	sylib 221	. . . . . . . 8 ⊢ (𝑠 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑠)) = 𝑠)
39	38	sseq1d 3961	. . . . . . 7 ⊢ (𝑠 ∈ 𝒫 𝐵 → ((𝐵 ∖ (𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
40	35, 39	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐵 ∖ (𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
41	34, 40	bitrd 282	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
42	5, 6	ntrclsbex 44978	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
43	42	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐵 ∈ V)
44	25	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
45		eqid 2760	. . . . . . 7 ⊢ (𝐷‘𝐼) = (𝐷‘𝐼)
46		eqid 2760	. . . . . . 7 ⊢ ((𝐷‘𝐼)‘𝑠) = ((𝐷‘𝐼)‘𝑠)
47	24, 5, 43, 44, 45, 35, 46	dssmapfv3d 44963	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐷‘𝐼)‘𝑠) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠))))
48	47	sseq2d 3962	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑠)))))
49	24, 5, 6	ntrclsfv1 44999	. . . . . . . 8 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾)
50	49	fveq1d 6875	. . . . . . 7 ⊢ (𝜑 → ((𝐷‘𝐼)‘𝑠) = (𝐾‘𝑠))
51	50	sseq2d 3962	. . . . . 6 ⊢ (𝜑 → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾‘𝑠)))
52	51	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑠 ⊆ ((𝐷‘𝐼)‘𝑠) ↔ 𝑠 ⊆ (𝐾‘𝑠)))
53	41, 48, 52	3bitr2d 310	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘(𝐵 ∖ 𝑠)) ⊆ (𝐵 ∖ 𝑠) ↔ 𝑠 ⊆ (𝐾‘𝑠)))
54	23, 53	bitrd 282	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → ((𝐼‘𝑡) ⊆ 𝑡 ↔ 𝑠 ⊆ (𝐾‘𝑠)))
55	8, 19, 54	ralxfrd2 5373	. 2 ⊢ (𝜑 → (∀𝑡 ∈ 𝒫 𝐵(𝐼‘𝑡) ⊆ 𝑡 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))
56	4, 55	bitrid 286	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘𝑠) ⊆ 𝑠 ↔ ∀𝑠 ∈ 𝒫 𝐵𝑠 ⊆ (𝐾‘𝑠)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ∖ cdif 3895   ⊆ wss 3898  𝒫 cpw 4556   class class class wbr 5102   ↦ cmpt 5185  ⟶wf 6523  ‘cfv 6527  (class class class)co 7408   ↑_m cmap 8825

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-map 8827

This theorem is used by: (None)