MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uzrest Structured version   Visualization version   GIF version

Theorem uzrest 21611
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 = (ℤ𝑀)
Assertion
Ref Expression
uzrest (𝑀 ∈ ℤ → (ran ℤt 𝑍) = (ℤ𝑍))

Proof of Theorem uzrest
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zex 11330 . . . . . 6 ℤ ∈ V
21pwex 4808 . . . . 5 𝒫 ℤ ∈ V
3 uzf 11634 . . . . . 6 :ℤ⟶𝒫 ℤ
4 frn 6010 . . . . . 6 (ℤ:ℤ⟶𝒫 ℤ → ran ℤ ⊆ 𝒫 ℤ)
53, 4ax-mp 5 . . . . 5 ran ℤ ⊆ 𝒫 ℤ
62, 5ssexi 4763 . . . 4 ran ℤ ∈ V
7 uzfbas.1 . . . . 5 𝑍 = (ℤ𝑀)
8 fvex 6158 . . . . 5 (ℤ𝑀) ∈ V
97, 8eqeltri 2694 . . . 4 𝑍 ∈ V
10 restval 16008 . . . 4 ((ran ℤ ∈ V ∧ 𝑍 ∈ V) → (ran ℤt 𝑍) = ran (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)))
116, 9, 10mp2an 707 . . 3 (ran ℤt 𝑍) = ran (𝑥 ∈ ran ℤ ↦ (𝑥𝑍))
127ineq2i 3789 . . . . . . . . 9 ((ℤ𝑦) ∩ 𝑍) = ((ℤ𝑦) ∩ (ℤ𝑀))
13 uzin 11664 . . . . . . . . . 10 ((𝑦 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((ℤ𝑦) ∩ (ℤ𝑀)) = (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
1413ancoms 469 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑦) ∩ (ℤ𝑀)) = (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
1512, 14syl5eq 2667 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑦) ∩ 𝑍) = (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
16 ffn 6002 . . . . . . . . . . 11 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
173, 16ax-mp 5 . . . . . . . . . 10 Fn ℤ
1817a1i 11 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ℤ Fn ℤ)
19 uzssz 11651 . . . . . . . . . . 11 (ℤ𝑀) ⊆ ℤ
207, 19eqsstri 3614 . . . . . . . . . 10 𝑍 ⊆ ℤ
2120a1i 11 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑍 ⊆ ℤ)
22 inss2 3812 . . . . . . . . . 10 ((ℤ𝑦) ∩ 𝑍) ⊆ 𝑍
23 ifcl 4102 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ ℤ)
24 uzid 11646 . . . . . . . . . . . 12 (if(𝑦𝑀, 𝑀, 𝑦) ∈ ℤ → if(𝑦𝑀, 𝑀, 𝑦) ∈ (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
2523, 24syl 17 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
2625, 15eleqtrrd 2701 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ ((ℤ𝑦) ∩ 𝑍))
2722, 26sseldi 3581 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ 𝑍)
28 fnfvima 6450 . . . . . . . . 9 ((ℤ Fn ℤ ∧ 𝑍 ⊆ ℤ ∧ if(𝑦𝑀, 𝑀, 𝑦) ∈ 𝑍) → (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)) ∈ (ℤ𝑍))
2918, 21, 27, 28syl3anc 1323 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)) ∈ (ℤ𝑍))
3015, 29eqeltrd 2698 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍))
3130ralrimiva 2960 . . . . . 6 (𝑀 ∈ ℤ → ∀𝑦 ∈ ℤ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍))
32 ineq1 3785 . . . . . . . . 9 (𝑥 = (ℤ𝑦) → (𝑥𝑍) = ((ℤ𝑦) ∩ 𝑍))
3332eleq1d 2683 . . . . . . . 8 (𝑥 = (ℤ𝑦) → ((𝑥𝑍) ∈ (ℤ𝑍) ↔ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍)))
3433ralrn 6318 . . . . . . 7 (ℤ Fn ℤ → (∀𝑥 ∈ ran ℤ(𝑥𝑍) ∈ (ℤ𝑍) ↔ ∀𝑦 ∈ ℤ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍)))
3517, 34ax-mp 5 . . . . . 6 (∀𝑥 ∈ ran ℤ(𝑥𝑍) ∈ (ℤ𝑍) ↔ ∀𝑦 ∈ ℤ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍))
3631, 35sylibr 224 . . . . 5 (𝑀 ∈ ℤ → ∀𝑥 ∈ ran ℤ(𝑥𝑍) ∈ (ℤ𝑍))
37 eqid 2621 . . . . . 6 (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)) = (𝑥 ∈ ran ℤ ↦ (𝑥𝑍))
3837fmpt 6337 . . . . 5 (∀𝑥 ∈ ran ℤ(𝑥𝑍) ∈ (ℤ𝑍) ↔ (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)):ran ℤ⟶(ℤ𝑍))
3936, 38sylib 208 . . . 4 (𝑀 ∈ ℤ → (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)):ran ℤ⟶(ℤ𝑍))
40 frn 6010 . . . 4 ((𝑥 ∈ ran ℤ ↦ (𝑥𝑍)):ran ℤ⟶(ℤ𝑍) → ran (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)) ⊆ (ℤ𝑍))
4139, 40syl 17 . . 3 (𝑀 ∈ ℤ → ran (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)) ⊆ (ℤ𝑍))
4211, 41syl5eqss 3628 . 2 (𝑀 ∈ ℤ → (ran ℤt 𝑍) ⊆ (ℤ𝑍))
437uztrn2 11649 . . . . . . . . 9 ((𝑥𝑍𝑦 ∈ (ℤ𝑥)) → 𝑦𝑍)
4443ex 450 . . . . . . . 8 (𝑥𝑍 → (𝑦 ∈ (ℤ𝑥) → 𝑦𝑍))
4544ssrdv 3589 . . . . . . 7 (𝑥𝑍 → (ℤ𝑥) ⊆ 𝑍)
4645adantl 482 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ⊆ 𝑍)
47 df-ss 3569 . . . . . 6 ((ℤ𝑥) ⊆ 𝑍 ↔ ((ℤ𝑥) ∩ 𝑍) = (ℤ𝑥))
4846, 47sylib 208 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → ((ℤ𝑥) ∩ 𝑍) = (ℤ𝑥))
496a1i 11 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → ran ℤ ∈ V)
509a1i 11 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → 𝑍 ∈ V)
5120sseli 3579 . . . . . . . 8 (𝑥𝑍𝑥 ∈ ℤ)
5251adantl 482 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → 𝑥 ∈ ℤ)
53 fnfvelrn 6312 . . . . . . 7 ((ℤ Fn ℤ ∧ 𝑥 ∈ ℤ) → (ℤ𝑥) ∈ ran ℤ)
5417, 52, 53sylancr 694 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ∈ ran ℤ)
55 elrestr 16010 . . . . . 6 ((ran ℤ ∈ V ∧ 𝑍 ∈ V ∧ (ℤ𝑥) ∈ ran ℤ) → ((ℤ𝑥) ∩ 𝑍) ∈ (ran ℤt 𝑍))
5649, 50, 54, 55syl3anc 1323 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → ((ℤ𝑥) ∩ 𝑍) ∈ (ran ℤt 𝑍))
5748, 56eqeltrrd 2699 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ∈ (ran ℤt 𝑍))
5857ralrimiva 2960 . . 3 (𝑀 ∈ ℤ → ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍))
59 ffun 6005 . . . . 5 (ℤ:ℤ⟶𝒫 ℤ → Fun ℤ)
603, 59ax-mp 5 . . . 4 Fun ℤ
613fdmi 6009 . . . . 5 dom ℤ = ℤ
6220, 61sseqtr4i 3617 . . . 4 𝑍 ⊆ dom ℤ
63 funimass4 6204 . . . 4 ((Fun ℤ𝑍 ⊆ dom ℤ) → ((ℤ𝑍) ⊆ (ran ℤt 𝑍) ↔ ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍)))
6460, 62, 63mp2an 707 . . 3 ((ℤ𝑍) ⊆ (ran ℤt 𝑍) ↔ ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍))
6558, 64sylibr 224 . 2 (𝑀 ∈ ℤ → (ℤ𝑍) ⊆ (ran ℤt 𝑍))
6642, 65eqssd 3600 1 (𝑀 ∈ ℤ → (ran ℤt 𝑍) = (ℤ𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cin 3554  wss 3555  ifcif 4058  𝒫 cpw 4130   class class class wbr 4613  cmpt 4673  dom cdm 5074  ran crn 5075  cima 5077  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  cle 10019  cz 11321  cuz 11631  t crest 16002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-neg 10213  df-z 11322  df-uz 11632  df-rest 16004
This theorem is referenced by:  uzfbas  21612
  Copyright terms: Public domain W3C validator