Theorem uzrest 23791

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 12545	. . . . . 6 ⊢ ℤ ∈ V
2	1	pwex 5338	. . . . 5 ⊢ 𝒫 ℤ ∈ V
3		uzf 12803	. . . . . 6 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
4		frn 6698	. . . . . 6 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ran ℤ≥ ⊆ 𝒫 ℤ)
5	3, 4	ax-mp 5	. . . . 5 ⊢ ran ℤ≥ ⊆ 𝒫 ℤ
6	2, 5	ssexi 5280	. . . 4 ⊢ ran ℤ≥ ∈ V
7		uzfbas.1	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
8	7	fvexi 6875	. . . 4 ⊢ 𝑍 ∈ V
9		restval 17396	. . . 4 ⊢ ((ran ℤ≥ ∈ V ∧ 𝑍 ∈ V) → (ran ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)))
10	6, 8, 9	mp2an 692	. . 3 ⊢ (ran ℤ≥ ↾t 𝑍) = ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍))
11	7	ineq2i 4183	. . . . . . . . 9 ⊢ ((ℤ≥‘𝑦) ∩ 𝑍) = ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀))
12		uzin 12840	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
13	12	ancoms 458	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ (ℤ≥‘𝑀)) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
14	11, 13	eqtrid 2777	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ 𝑍) = (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)))
15		ffn 6691	. . . . . . . . . 10 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
16	3, 15	ax-mp 5	. . . . . . . . 9 ⊢ ℤ≥ Fn ℤ
17		uzssz 12821	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
18	7, 17	eqsstri 3996	. . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ
19		ifcl 4537	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ℤ)
20		uzid 12815	. . . . . . . . . . . 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 2832	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ ((ℤ≥‘𝑦) ∩ 𝑍))
23	22	elin2d 4171	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦 ≤ 𝑀, 𝑀, 𝑦) ∈ 𝑍)
24		fnfvima 7210	. . . . . . . . 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 1467	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ≥‘if(𝑦 ≤ 𝑀, 𝑀, 𝑦)) ∈ (ℤ≥ “ 𝑍))
26	14, 25	eqeltrd 2829	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
27	26	ralrimiva 3126	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ∀𝑦 ∈ ℤ ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
28		ineq1 4179	. . . . . . . . 9 ⊢ (𝑥 = (ℤ≥‘𝑦) → (𝑥 ∩ 𝑍) = ((ℤ≥‘𝑦) ∩ 𝑍))
29	28	eleq1d 2814	. . . . . . . 8 ⊢ (𝑥 = (ℤ≥‘𝑦) → ((𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍) ↔ ((ℤ≥‘𝑦) ∩ 𝑍) ∈ (ℤ≥ “ 𝑍)))
30	29	ralrn 7063	. . . . . . 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 234	. . . . 5 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍))
33		eqid 2730	. . . . . 6 ⊢ (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)) = (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍))
34	33	fmpt 7085	. . . . 5 ⊢ (∀𝑥 ∈ ran ℤ≥(𝑥 ∩ 𝑍) ∈ (ℤ≥ “ 𝑍) ↔ (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran ℤ≥⟶(ℤ≥ “ 𝑍))
35	32, 34	sylib 218	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)):ran ℤ≥⟶(ℤ≥ “ 𝑍))
36	35	frnd 6699	. . 3 ⊢ (𝑀 ∈ ℤ → ran (𝑥 ∈ ran ℤ≥ ↦ (𝑥 ∩ 𝑍)) ⊆ (ℤ≥ “ 𝑍))
37	10, 36	eqsstrid 3988	. 2 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) ⊆ (ℤ≥ “ 𝑍))
38	7	uztrn2 12819	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑍 ∧ 𝑦 ∈ (ℤ≥‘𝑥)) → 𝑦 ∈ 𝑍)
39	38	ex 412	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑍 → (𝑦 ∈ (ℤ≥‘𝑥) → 𝑦 ∈ 𝑍))
40	39	ssrdv 3955	. . . . . . 7 ⊢ (𝑥 ∈ 𝑍 → (ℤ≥‘𝑥) ⊆ 𝑍)
41	40	adantl 481	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ⊆ 𝑍)
42		dfss2 3935	. . . . . 6 ⊢ ((ℤ≥‘𝑥) ⊆ 𝑍 ↔ ((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥))
43	41, 42	sylib 218	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → ((ℤ≥‘𝑥) ∩ 𝑍) = (ℤ≥‘𝑥))
44	18	sseli 3945	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑍 → 𝑥 ∈ ℤ)
45	44	adantl 481	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ ℤ)
46		fnfvelrn 7055	. . . . . . 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 17398	. . . . . 6 ⊢ ((ran ℤ≥ ∈ V ∧ 𝑍 ∈ V ∧ (ℤ≥‘𝑥) ∈ ran ℤ≥) → ((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥ ↾t 𝑍))
49	6, 8, 47, 48	mp3an12i 1467	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → ((ℤ≥‘𝑥) ∩ 𝑍) ∈ (ran ℤ≥ ↾t 𝑍))
50	43, 49	eqeltrrd 2830	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑥 ∈ 𝑍) → (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
51	50	ralrimiva 3126	. . 3 ⊢ (𝑀 ∈ ℤ → ∀𝑥 ∈ 𝑍 (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
52		ffun 6694	. . . . 5 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → Fun ℤ≥)
53	3, 52	ax-mp 5	. . . 4 ⊢ Fun ℤ≥
54	3	fdmi 6702	. . . . 5 ⊢ dom ℤ≥ = ℤ
55	18, 54	sseqtrri 3999	. . . 4 ⊢ 𝑍 ⊆ dom ℤ≥
56		funimass4 6928	. . . 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 692	. . 3 ⊢ ((ℤ≥ “ 𝑍) ⊆ (ran ℤ≥ ↾t 𝑍) ↔ ∀𝑥 ∈ 𝑍 (ℤ≥‘𝑥) ∈ (ran ℤ≥ ↾t 𝑍))
58	51, 57	sylibr 234	. 2 ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ⊆ (ran ℤ≥ ↾t 𝑍))
59	37, 58	eqssd 3967	1 ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) = (ℤ≥ “ 𝑍))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917  ifcif 4491  𝒫 cpw 4566   class class class wbr 5110   ↦ cmpt 5191  dom cdm 5641  ran crn 5642   “ cima 5644  Fun wfun 6508   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ≤ cle 11216  ℤcz 12536  ℤ_≥cuz 12800   ↾_t crest 17390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-pre-lttri 11149  ax-pre-lttrn 11150

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-neg 11415  df-z 12537  df-uz 12801  df-rest 17392

This theorem is referenced by:  uzfbas  23792