Theorem uzrest 22499

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 11984	. . . . . 6 ⊢ ℤ ∈ V
2	1	pwex 5273	. . . . 5 ⊢ 𝒫 ℤ ∈ V
3		uzf 12240	. . . . . 6 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
4		frn 6514	. . . . . 6 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ran ℤ≥ ⊆ 𝒫 ℤ)
5	3, 4	ax-mp 5	. . . . 5 ⊢ ran ℤ≥ ⊆ 𝒫 ℤ
6	2, 5	ssexi 5218	. . . 4 ⊢ ran ℤ≥ ∈ V
7		uzfbas.1	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
8	7	fvexi 6678	. . . 4 ⊢ 𝑍 ∈ V
9		restval 16694	. . . 4 ⊢ ((ran ℤ≥ ∈ V ∧ 𝑍 ∈ V) → (ran ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)))
10	6, 8, 9	mp2an 690	. . 3 ⊢ (ran ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍))
11	7	ineq2i 4185	. . . . . . . . 9 ⊢ ((ℤ≥‘𝑦) ∩ 𝑍) = ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀))
12		uzin 12272	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
13	12	ancoms 461	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
14	11, 13	syl5eq 2868	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ 𝑍) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
15		ffn 6508	. . . . . . . . . 10 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
16	3, 15	ax-mp 5	. . . . . . . . 9 ⊢ ℤ≥ Fn ℤ
17		uzssz 12258	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
18	7, 17	eqsstri 4000	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ
19		ifcl 4510	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ℤ)
20		uzid 12252	. . . . . . . . . . . 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 2916	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ((ℤ≥‘𝑦) ∩ 𝑍))
23	22	elin2d 4175	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ 𝑍)
24		fnfvima 6989	. . . . . . . . 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 1461	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)) ∈ (ℤ≥ “ 𝑍))
26	14, 25	eqeltrd 2913	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
27	26	ralrimiva 3182	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ∀𝑦 ∈ ℤ ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
28		ineq1 4180	. . . . . . . . 9 ⊢ (𝑥 = (ℤ≥‘𝑦) → (𝑥 ∩ 𝑍) = ((ℤ≥‘𝑦) ∩ 𝑍))
29	28	eleq1d 2897	. . . . . . . 8 ⊢ (𝑥 = (ℤ≥‘𝑦) → ((𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍) ↔ ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍)))
30	29	ralrn 6848	. . . . . . 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 236	. . . . 5 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
33		eqid 2821	. . . . . 6 ⊢ (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)) = (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍))
34	33	fmpt 6868	. . . . 5 ⊢ (∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍) ↔ (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran ℤ≥⟶(ℤ≥ “ 𝑍))
35	32, 34	sylib 220	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran ℤ≥⟶(ℤ≥ “ 𝑍))
36	35	frnd 6515	. . 3 ⊢ (𝑀 ∈ ℤ → ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)) ⊆ (ℤ≥ “ 𝑍))
37	10, 36	eqsstrid 4014	. 2 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) ⊆ (ℤ≥ “ 𝑍))
38	7	uztrn2 12256	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑍 ∧ 𝑦 ∈ (ℤ≥‘𝑥)) → 𝑦 ∈ 𝑍)
39	38	ex 415	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑍 → (𝑦 ∈ (ℤ≥‘𝑥) → 𝑦 ∈ 𝑍))
40	39	ssrdv 3972	. . . . . . 7 ⊢ (𝑥 ∈ 𝑍 → (ℤ≥‘𝑥) ⊆ 𝑍)
41	40	adantl 484	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ⊆ 𝑍)
42		df-ss 3951	. . . . . 6 ⊢ ((ℤ≥‘𝑥) ⊆ 𝑍 ↔ ((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥))
43	41, 42	sylib 220	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → ((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥))
44	18	sseli 3962	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑍 → 𝑥 ∈ ℤ)
45	44	adantl 484	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ ℤ)
46		fnfvelrn 6842	. . . . . . 7 ⊢ ((ℤ≥ Fn ℤ ∧ 𝑥 ∈ ℤ) → (ℤ≥‘𝑥) ∈ ran ℤ≥)
47	16, 45, 46	sylancr 589	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ∈ ran ℤ≥)
48		elrestr 16696	. . . . . 6 ⊢ ((ran ℤ≥ ∈ V ∧ 𝑍 ∈ V ∧ (ℤ≥‘𝑥) ∈ ran ℤ≥) → ((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥ ↾t 𝑍))
49	6, 8, 47, 48	mp3an12i 1461	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → ((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥ ↾t 𝑍))
50	43, 49	eqeltrrd 2914	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
51	50	ralrimiva 3182	. . 3 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ 𝑍 (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
52		ffun 6511	. . . . 5 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → Fun ℤ≥)
53	3, 52	ax-mp 5	. . . 4 ⊢ Fun ℤ≥
54	3	fdmi 6518	. . . . 5 ⊢ dom ℤ≥ = ℤ
55	18, 54	sseqtrri 4003	. . . 4 ⊢ 𝑍 ⊆ dom ℤ≥
56		funimass4 6724	. . . 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 690	. . 3 ⊢ ((ℤ≥ “ 𝑍) ⊆ (ran ℤ≥ ↾t 𝑍) ↔ ∀𝑥 ∈ 𝑍 (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
58	51, 57	sylibr 236	. 2 ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ⊆ (ran ℤ≥ ↾t 𝑍))
59	37, 58	eqssd 3983	1 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) = (ℤ≥ “ 𝑍))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494   ∩ cin 3934   ⊆ wss 3935  ifcif 4466  𝒫 cpw 4538   class class class wbr 5058   ↦ cmpt 5138  dom cdm 5549  ran crn 5550   “ cima 5552  Fun wfun 6343   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ≤ cle 10670  ℤcz 11975  ℤ_≥cuz 12237   ↾_t crest 16688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-pre-lttri 10605  ax-pre-lttrn 10606

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-po 5468  df-so 5469  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-neg 10867  df-z 11976  df-uz 12238  df-rest 16690

This theorem is referenced by:  uzfbas  22500