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Theorem uzrest 21748
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 11424 . . . . . 6 ℤ ∈ V
21pwex 4878 . . . . 5 𝒫 ℤ ∈ V
3 uzf 11728 . . . . . 6 :ℤ⟶𝒫 ℤ
4 frn 6091 . . . . . 6 (ℤ:ℤ⟶𝒫 ℤ → ran ℤ ⊆ 𝒫 ℤ)
53, 4ax-mp 5 . . . . 5 ran ℤ ⊆ 𝒫 ℤ
62, 5ssexi 4836 . . . 4 ran ℤ ∈ V
7 uzfbas.1 . . . . 5 𝑍 = (ℤ𝑀)
8 fvex 6239 . . . . 5 (ℤ𝑀) ∈ V
97, 8eqeltri 2726 . . . 4 𝑍 ∈ V
10 restval 16134 . . . 4 ((ran ℤ ∈ V ∧ 𝑍 ∈ V) → (ran ℤt 𝑍) = ran (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)))
116, 9, 10mp2an 708 . . 3 (ran ℤt 𝑍) = ran (𝑥 ∈ ran ℤ ↦ (𝑥𝑍))
127ineq2i 3844 . . . . . . . . 9 ((ℤ𝑦) ∩ 𝑍) = ((ℤ𝑦) ∩ (ℤ𝑀))
13 uzin 11758 . . . . . . . . . 10 ((𝑦 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((ℤ𝑦) ∩ (ℤ𝑀)) = (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
1413ancoms 468 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑦) ∩ (ℤ𝑀)) = (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
1512, 14syl5eq 2697 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑦) ∩ 𝑍) = (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
16 ffn 6083 . . . . . . . . . . 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 11745 . . . . . . . . . . 11 (ℤ𝑀) ⊆ ℤ
207, 19eqsstri 3668 . . . . . . . . . 10 𝑍 ⊆ ℤ
2120a1i 11 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑍 ⊆ ℤ)
22 inss2 3867 . . . . . . . . . 10 ((ℤ𝑦) ∩ 𝑍) ⊆ 𝑍
23 ifcl 4163 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ ℤ)
24 uzid 11740 . . . . . . . . . . . 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 2733 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ ((ℤ𝑦) ∩ 𝑍))
2722, 26sseldi 3634 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ 𝑍)
28 fnfvima 6536 . . . . . . . . 9 ((ℤ Fn ℤ ∧ 𝑍 ⊆ ℤ ∧ if(𝑦𝑀, 𝑀, 𝑦) ∈ 𝑍) → (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)) ∈ (ℤ𝑍))
2918, 21, 27, 28syl3anc 1366 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)) ∈ (ℤ𝑍))
3015, 29eqeltrd 2730 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍))
3130ralrimiva 2995 . . . . . 6 (𝑀 ∈ ℤ → ∀𝑦 ∈ ℤ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍))
32 ineq1 3840 . . . . . . . . 9 (𝑥 = (ℤ𝑦) → (𝑥𝑍) = ((ℤ𝑦) ∩ 𝑍))
3332eleq1d 2715 . . . . . . . 8 (𝑥 = (ℤ𝑦) → ((𝑥𝑍) ∈ (ℤ𝑍) ↔ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍)))
3433ralrn 6402 . . . . . . 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 2651 . . . . . 6 (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)) = (𝑥 ∈ ran ℤ ↦ (𝑥𝑍))
3837fmpt 6421 . . . . 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 6091 . . . 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 3682 . 2 (𝑀 ∈ ℤ → (ran ℤt 𝑍) ⊆ (ℤ𝑍))
437uztrn2 11743 . . . . . . . . 9 ((𝑥𝑍𝑦 ∈ (ℤ𝑥)) → 𝑦𝑍)
4443ex 449 . . . . . . . 8 (𝑥𝑍 → (𝑦 ∈ (ℤ𝑥) → 𝑦𝑍))
4544ssrdv 3642 . . . . . . 7 (𝑥𝑍 → (ℤ𝑥) ⊆ 𝑍)
4645adantl 481 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ⊆ 𝑍)
47 df-ss 3621 . . . . . 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 3632 . . . . . . . 8 (𝑥𝑍𝑥 ∈ ℤ)
5251adantl 481 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → 𝑥 ∈ ℤ)
53 fnfvelrn 6396 . . . . . . 7 ((ℤ Fn ℤ ∧ 𝑥 ∈ ℤ) → (ℤ𝑥) ∈ ran ℤ)
5417, 52, 53sylancr 696 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ∈ ran ℤ)
55 elrestr 16136 . . . . . 6 ((ran ℤ ∈ V ∧ 𝑍 ∈ V ∧ (ℤ𝑥) ∈ ran ℤ) → ((ℤ𝑥) ∩ 𝑍) ∈ (ran ℤt 𝑍))
5649, 50, 54, 55syl3anc 1366 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → ((ℤ𝑥) ∩ 𝑍) ∈ (ran ℤt 𝑍))
5748, 56eqeltrrd 2731 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ∈ (ran ℤt 𝑍))
5857ralrimiva 2995 . . 3 (𝑀 ∈ ℤ → ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍))
59 ffun 6086 . . . . 5 (ℤ:ℤ⟶𝒫 ℤ → Fun ℤ)
603, 59ax-mp 5 . . . 4 Fun ℤ
613fdmi 6090 . . . . 5 dom ℤ = ℤ
6220, 61sseqtr4i 3671 . . . 4 𝑍 ⊆ dom ℤ
63 funimass4 6286 . . . 4 ((Fun ℤ𝑍 ⊆ dom ℤ) → ((ℤ𝑍) ⊆ (ran ℤt 𝑍) ↔ ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍)))
6460, 62, 63mp2an 708 . . 3 ((ℤ𝑍) ⊆ (ran ℤt 𝑍) ↔ ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍))
6558, 64sylibr 224 . 2 (𝑀 ∈ ℤ → (ℤ𝑍) ⊆ (ran ℤt 𝑍))
6642, 65eqssd 3653 1 (𝑀 ∈ ℤ → (ran ℤt 𝑍) = (ℤ𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cin 3606  wss 3607  ifcif 4119  𝒫 cpw 4191   class class class wbr 4685  cmpt 4762  dom cdm 5143  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  cle 10113  cz 11415  cuz 11725  t crest 16128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-neg 10307  df-z 11416  df-uz 11726  df-rest 16130
This theorem is referenced by:  uzfbas  21749
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