Theorem uzrest 23048

Description: The restriction of the set of upper sets of integers to an upper set of integers is the set of upper sets of integers based at a point above the cutoff. (Contributed by Mario Carneiro, 13-Oct-2015.)

Hypothesis

Ref	Expression
uzfbas.1	⊢ 𝑍 = (ℤ≥‘𝑀)

Assertion

Ref	Expression
uzrest	⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) = (ℤ≥ “ 𝑍))

Proof of Theorem uzrest

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		zex 12328	. . . . . 6 ⊢ ℤ ∈ V
2	1	pwex 5303	. . . . 5 ⊢ 𝒫 ℤ ∈ V
3		uzf 12585	. . . . . 6 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
4		frn 6607	. . . . . 6 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ran ℤ≥ ⊆ 𝒫 ℤ)
5	3, 4	ax-mp 5	. . . . 5 ⊢ ran ℤ≥ ⊆ 𝒫 ℤ
6	2, 5	ssexi 5246	. . . 4 ⊢ ran ℤ≥ ∈ V
7		uzfbas.1	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
8	7	fvexi 6788	. . . 4 ⊢ 𝑍 ∈ V
9		restval 17137	. . . 4 ⊢ ((ran ℤ≥ ∈ V ∧ 𝑍 ∈ V) → (ran ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)))
10	6, 8, 9	mp2an 689	. . 3 ⊢ (ran ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍))
11	7	ineq2i 4143	. . . . . . . . 9 ⊢ ((ℤ≥‘𝑦) ∩ 𝑍) = ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀))
12		uzin 12618	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
13	12	ancoms 459	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
14	11, 13	eqtrid 2790	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ 𝑍) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
15		ffn 6600	. . . . . . . . . 10 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
16	3, 15	ax-mp 5	. . . . . . . . 9 ⊢ ℤ≥ Fn ℤ
17		uzssz 12603	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
18	7, 17	eqsstri 3955	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ
19		ifcl 4504	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ℤ)
20		uzid 12597	. . . . . . . . . . . 12 ⊢ (if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ℤ → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
22	21, 14	eleqtrrd 2842	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ((ℤ≥‘𝑦) ∩ 𝑍))
23	22	elin2d 4133	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ 𝑍)
24		fnfvima 7109	. . . . . . . . 9 ⊢ ((ℤ≥ Fn ℤ ∧ 𝑍 ⊆ ℤ ∧ if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ 𝑍) → (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)) ∈ (ℤ≥ “ 𝑍))
25	16, 18, 23, 24	mp3an12i 1464	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)) ∈ (ℤ≥ “ 𝑍))
26	14, 25	eqeltrd 2839	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
27	26	ralrimiva 3103	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ∀𝑦 ∈ ℤ ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
28		ineq1 4139	. . . . . . . . 9 ⊢ (𝑥 = (ℤ≥‘𝑦) → (𝑥 ∩ 𝑍) = ((ℤ≥‘𝑦) ∩ 𝑍))
29	28	eleq1d 2823	. . . . . . . 8 ⊢ (𝑥 = (ℤ≥‘𝑦) → ((𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍) ↔ ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍)))
30	29	ralrn 6964	. . . . . . 7 ⊢ (ℤ≥ Fn ℤ → (∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍) ↔ ∀𝑦 ∈ ℤ ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍)))
31	16, 30	ax-mp 5	. . . . . 6 ⊢ (∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍) ↔ ∀𝑦 ∈ ℤ ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
32	27, 31	sylibr 233	. . . . 5 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
33		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)) = (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍))
34	33	fmpt 6984	. . . . 5 ⊢ (∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍) ↔ (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran ℤ≥⟶(ℤ≥ “ 𝑍))
35	32, 34	sylib 217	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran ℤ≥⟶(ℤ≥ “ 𝑍))
36	35	frnd 6608	. . 3 ⊢ (𝑀 ∈ ℤ → ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)) ⊆ (ℤ≥ “ 𝑍))
37	10, 36	eqsstrid 3969	. 2 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) ⊆ (ℤ≥ “ 𝑍))
38	7	uztrn2 12601	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑍 ∧ 𝑦 ∈ (ℤ≥‘𝑥)) → 𝑦 ∈ 𝑍)
39	38	ex 413	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑍 → (𝑦 ∈ (ℤ≥‘𝑥) → 𝑦 ∈ 𝑍))
40	39	ssrdv 3927	. . . . . . 7 ⊢ (𝑥 ∈ 𝑍 → (ℤ≥‘𝑥) ⊆ 𝑍)
41	40	adantl 482	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ⊆ 𝑍)
42		df-ss 3904	. . . . . 6 ⊢ ((ℤ≥‘𝑥) ⊆ 𝑍 ↔ ((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥))
43	41, 42	sylib 217	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → ((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥))
44	18	sseli 3917	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑍 → 𝑥 ∈ ℤ)
45	44	adantl 482	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ ℤ)
46		fnfvelrn 6958	. . . . . . 7 ⊢ ((ℤ≥ Fn ℤ ∧ 𝑥 ∈ ℤ) → (ℤ≥‘𝑥) ∈ ran ℤ≥)
47	16, 45, 46	sylancr 587	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ∈ ran ℤ≥)
48		elrestr 17139	. . . . . 6 ⊢ ((ran ℤ≥ ∈ V ∧ 𝑍 ∈ V ∧ (ℤ≥‘𝑥) ∈ ran ℤ≥) → ((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥ ↾t 𝑍))
49	6, 8, 47, 48	mp3an12i 1464	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → ((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥ ↾t 𝑍))
50	43, 49	eqeltrrd 2840	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
51	50	ralrimiva 3103	. . 3 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ 𝑍 (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
52		ffun 6603	. . . . 5 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → Fun ℤ≥)
53	3, 52	ax-mp 5	. . . 4 ⊢ Fun ℤ≥
54	3	fdmi 6612	. . . . 5 ⊢ dom ℤ≥ = ℤ
55	18, 54	sseqtrri 3958	. . . 4 ⊢ 𝑍 ⊆ dom ℤ≥
56		funimass4 6834	. . . 4 ⊢ ((Fun ℤ≥ ∧ 𝑍 ⊆ dom ℤ≥) → ((ℤ≥ “ 𝑍) ⊆ (ran ℤ≥ ↾t 𝑍) ↔ ∀𝑥 ∈ 𝑍 (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍)))
57	53, 55, 56	mp2an 689	. . 3 ⊢ ((ℤ≥ “ 𝑍) ⊆ (ran ℤ≥ ↾t 𝑍) ↔ ∀𝑥 ∈ 𝑍 (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
58	51, 57	sylibr 233	. 2 ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ⊆ (ran ℤ≥ ↾t 𝑍))
59	37, 58	eqssd 3938	1 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) = (ℤ≥ “ 𝑍))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ifcif 4459  𝒫 cpw 4533   class class class wbr 5074   ↦ cmpt 5157  dom cdm 5589  ran crn 5590   “ cima 5592  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ≤ cle 11010  ℤcz 12319  ℤ_≥cuz 12582   ↾_t crest 17131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-pre-lttri 10945  ax-pre-lttrn 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-neg 11208  df-z 12320  df-uz 12583  df-rest 17133

This theorem is referenced by:  uzfbas  23049