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Theorem uzrest 21922
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 11598 . . . . . 6 ℤ ∈ V
21pwex 4997 . . . . 5 𝒫 ℤ ∈ V
3 uzf 11902 . . . . . 6 :ℤ⟶𝒫 ℤ
4 frn 6214 . . . . . 6 (ℤ:ℤ⟶𝒫 ℤ → ran ℤ ⊆ 𝒫 ℤ)
53, 4ax-mp 5 . . . . 5 ran ℤ ⊆ 𝒫 ℤ
62, 5ssexi 4955 . . . 4 ran ℤ ∈ V
7 uzfbas.1 . . . . 5 𝑍 = (ℤ𝑀)
8 fvex 6363 . . . . 5 (ℤ𝑀) ∈ V
97, 8eqeltri 2835 . . . 4 𝑍 ∈ V
10 restval 16309 . . . 4 ((ran ℤ ∈ V ∧ 𝑍 ∈ V) → (ran ℤt 𝑍) = ran (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)))
116, 9, 10mp2an 710 . . 3 (ran ℤt 𝑍) = ran (𝑥 ∈ ran ℤ ↦ (𝑥𝑍))
127ineq2i 3954 . . . . . . . . 9 ((ℤ𝑦) ∩ 𝑍) = ((ℤ𝑦) ∩ (ℤ𝑀))
13 uzin 11933 . . . . . . . . . 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 2806 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑦) ∩ 𝑍) = (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
16 ffn 6206 . . . . . . . . . . 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 11919 . . . . . . . . . . 11 (ℤ𝑀) ⊆ ℤ
207, 19eqsstri 3776 . . . . . . . . . 10 𝑍 ⊆ ℤ
2120a1i 11 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑍 ⊆ ℤ)
22 inss2 3977 . . . . . . . . . 10 ((ℤ𝑦) ∩ 𝑍) ⊆ 𝑍
23 ifcl 4274 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ ℤ)
24 uzid 11914 . . . . . . . . . . . 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 2842 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ ((ℤ𝑦) ∩ 𝑍))
2722, 26sseldi 3742 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ 𝑍)
28 fnfvima 6660 . . . . . . . . 9 ((ℤ Fn ℤ ∧ 𝑍 ⊆ ℤ ∧ if(𝑦𝑀, 𝑀, 𝑦) ∈ 𝑍) → (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)) ∈ (ℤ𝑍))
2918, 21, 27, 28syl3anc 1477 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)) ∈ (ℤ𝑍))
3015, 29eqeltrd 2839 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍))
3130ralrimiva 3104 . . . . . 6 (𝑀 ∈ ℤ → ∀𝑦 ∈ ℤ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍))
32 ineq1 3950 . . . . . . . . 9 (𝑥 = (ℤ𝑦) → (𝑥𝑍) = ((ℤ𝑦) ∩ 𝑍))
3332eleq1d 2824 . . . . . . . 8 (𝑥 = (ℤ𝑦) → ((𝑥𝑍) ∈ (ℤ𝑍) ↔ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍)))
3433ralrn 6526 . . . . . . 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 2760 . . . . . 6 (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)) = (𝑥 ∈ ran ℤ ↦ (𝑥𝑍))
3837fmpt 6545 . . . . 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 6214 . . . 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 3790 . 2 (𝑀 ∈ ℤ → (ran ℤt 𝑍) ⊆ (ℤ𝑍))
437uztrn2 11917 . . . . . . . . 9 ((𝑥𝑍𝑦 ∈ (ℤ𝑥)) → 𝑦𝑍)
4443ex 449 . . . . . . . 8 (𝑥𝑍 → (𝑦 ∈ (ℤ𝑥) → 𝑦𝑍))
4544ssrdv 3750 . . . . . . 7 (𝑥𝑍 → (ℤ𝑥) ⊆ 𝑍)
4645adantl 473 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ⊆ 𝑍)
47 df-ss 3729 . . . . . 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 3740 . . . . . . . 8 (𝑥𝑍𝑥 ∈ ℤ)
5251adantl 473 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → 𝑥 ∈ ℤ)
53 fnfvelrn 6520 . . . . . . 7 ((ℤ Fn ℤ ∧ 𝑥 ∈ ℤ) → (ℤ𝑥) ∈ ran ℤ)
5417, 52, 53sylancr 698 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ∈ ran ℤ)
55 elrestr 16311 . . . . . 6 ((ran ℤ ∈ V ∧ 𝑍 ∈ V ∧ (ℤ𝑥) ∈ ran ℤ) → ((ℤ𝑥) ∩ 𝑍) ∈ (ran ℤt 𝑍))
5649, 50, 54, 55syl3anc 1477 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → ((ℤ𝑥) ∩ 𝑍) ∈ (ran ℤt 𝑍))
5748, 56eqeltrrd 2840 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ∈ (ran ℤt 𝑍))
5857ralrimiva 3104 . . 3 (𝑀 ∈ ℤ → ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍))
59 ffun 6209 . . . . 5 (ℤ:ℤ⟶𝒫 ℤ → Fun ℤ)
603, 59ax-mp 5 . . . 4 Fun ℤ
613fdmi 6213 . . . . 5 dom ℤ = ℤ
6220, 61sseqtr4i 3779 . . . 4 𝑍 ⊆ dom ℤ
63 funimass4 6410 . . . 4 ((Fun ℤ𝑍 ⊆ dom ℤ) → ((ℤ𝑍) ⊆ (ran ℤt 𝑍) ↔ ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍)))
6460, 62, 63mp2an 710 . . 3 ((ℤ𝑍) ⊆ (ran ℤt 𝑍) ↔ ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍))
6558, 64sylibr 224 . 2 (𝑀 ∈ ℤ → (ℤ𝑍) ⊆ (ran ℤt 𝑍))
6642, 65eqssd 3761 1 (𝑀 ∈ ℤ → (ran ℤt 𝑍) = (ℤ𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cin 3714  wss 3715  ifcif 4230  𝒫 cpw 4302   class class class wbr 4804  cmpt 4881  dom cdm 5266  ran crn 5267  cima 5269  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  cle 10287  cz 11589  cuz 11899  t crest 16303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-pre-lttri 10222  ax-pre-lttrn 10223
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-neg 10481  df-z 11590  df-uz 11900  df-rest 16305
This theorem is referenced by:  uzfbas  21923
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