Theorem uzrest 24054

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 12595	. . . . . 6 ⊢ ℤ ∈ V
2	1	pwex 5351	. . . . 5 ⊢ 𝒫 ℤ ∈ V
3		uzf 12860	. . . . . 6 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
4		frn 6713	. . . . . 6 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ran ℤ≥ ⊆ 𝒫 ℤ)
5	3, 4	ax-mp 5	. . . . 5 ⊢ ran ℤ≥ ⊆ 𝒫 ℤ
6	2, 5	ssexi 5293	. . . 4 ⊢ ran ℤ≥ ∈ V
7		uzfbas.1	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
8	7	fvexi 6895	. . . 4 ⊢ 𝑍 ∈ V
9		restval 17474	. . . 4 ⊢ ((ran ℤ≥ ∈ V ∧ 𝑍 ∈ V) → (ran ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)))
10	6, 8, 9	mp2an 704	. . 3 ⊢ (ran ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍))
11	7	ineq2i 4170	. . . . . . . . 9 ⊢ ((ℤ≥‘𝑦) ∩ 𝑍) = ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀))
12		uzin 12893	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
13	12	ancoms 463	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
14	11, 13	eqtrid 2810	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ 𝑍) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
15		ffn 6705	. . . . . . . . . 10 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
16	3, 15	ax-mp 5	. . . . . . . . 9 ⊢ ℤ≥ Fn ℤ
17		uzssz 12878	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
18	7, 17	eqsstri 3983	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ
19		ifcl 4533	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ℤ)
20		uzid 12872	. . . . . . . . . . . 12 ⊢ (if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ℤ → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
21	19, 20	syl 18	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
22	21, 14	eleqtrrd 2866	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ((ℤ≥‘𝑦) ∩ 𝑍))
23	22	elin2d 4158	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ 𝑍)
24		fnfvima 7231	. . . . . . . . 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 1494	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)) ∈ (ℤ≥ “ 𝑍))
26	14, 25	eqeltrd 2863	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
27	26	ralrimiva 3157	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ∀𝑦 ∈ ℤ ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
28		ineq1 4166	. . . . . . . . 9 ⊢ (𝑥 = (ℤ≥‘𝑦) → (𝑥 ∩ 𝑍) = ((ℤ≥‘𝑦) ∩ 𝑍))
29	28	eleq1d 2848	. . . . . . . 8 ⊢ (𝑥 = (ℤ≥‘𝑦) → ((𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍) ↔ ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍)))
30	29	ralrn 7083	. . . . . . 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 237	. . . . 5 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
33		eqid 2763	. . . . . 6 ⊢ (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)) = (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍))
34	33	fmpt 7105	. . . . 5 ⊢ (∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍) ↔ (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran ℤ≥⟶(ℤ≥ “ 𝑍))
35	32, 34	sylib 221	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran ℤ≥⟶(ℤ≥ “ 𝑍))
36	35	frnd 6714	. . 3 ⊢ (𝑀 ∈ ℤ → ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)) ⊆ (ℤ≥ “ 𝑍))
37	10, 36	eqsstrid 3975	. 2 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) ⊆ (ℤ≥ “ 𝑍))
38	7	uztrn2 12876	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑍 ∧ 𝑦 ∈ (ℤ≥‘𝑥)) → 𝑦 ∈ 𝑍)
39	38	ex 417	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑍 → (𝑦 ∈ (ℤ≥‘𝑥) → 𝑦 ∈ 𝑍))
40	39	ssrdv 3943	. . . . . . 7 ⊢ (𝑥 ∈ 𝑍 → (ℤ≥‘𝑥) ⊆ 𝑍)
41	40	adantl 486	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ⊆ 𝑍)
42		dfss2 3923	. . . . . 6 ⊢ ((ℤ≥‘𝑥) ⊆ 𝑍 ↔ ((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥))
43	41, 42	sylib 221	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → ((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥))
44	18	sseli 3933	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑍 → 𝑥 ∈ ℤ)
45	44	adantl 486	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ ℤ)
46		fnfvelrn 7075	. . . . . . 7 ⊢ ((ℤ≥ Fn ℤ ∧ 𝑥 ∈ ℤ) → (ℤ≥‘𝑥) ∈ ran ℤ≥)
47	16, 45, 46	sylancr 598	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ∈ ran ℤ≥)
48		elrestr 17476	. . . . . 6 ⊢ ((ran ℤ≥ ∈ V ∧ 𝑍 ∈ V ∧ (ℤ≥‘𝑥) ∈ ran ℤ≥) → ((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥ ↾t 𝑍))
49	6, 8, 47, 48	mp3an12i 1494	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → ((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥ ↾t 𝑍))
50	43, 49	eqeltrrd 2864	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
51	50	ralrimiva 3157	. . 3 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ 𝑍 (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
52		ffun 6708	. . . . 5 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → Fun ℤ≥)
53	3, 52	ax-mp 5	. . . 4 ⊢ Fun ℤ≥
54	3	fdmi 6717	. . . . 5 ⊢ dom ℤ≥ = ℤ
55	18, 54	sseqtrri 3986	. . . 4 ⊢ 𝑍 ⊆ dom ℤ≥
56		funimass4 6945	. . . 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 704	. . 3 ⊢ ((ℤ≥ “ 𝑍) ⊆ (ran ℤ≥ ↾t 𝑍) ↔ ∀𝑥 ∈ 𝑍 (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
58	51, 57	sylibr 237	. 2 ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ⊆ (ran ℤ≥ ↾t 𝑍))
59	37, 58	eqssd 3954	1 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) = (ℤ≥ “ 𝑍))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ifcif 4487  𝒫 cpw 4562   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5661  ran crn 5662   “ cima 5664  Fun wfun 6530   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ≤ cle 11239  ℤcz 12586  ℤ_≥cuz 12857   ↾_t crest 17468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-pre-lttri 11169  ax-pre-lttrn 11170

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-po 5569  df-so 5570  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-neg 11439  df-z 12587  df-uz 12858  df-rest 17470

This theorem is referenced by:  uzfbas  24055