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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  ℤcz 11415  ℤ≥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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