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Theorem uzrest 23248
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 12508 . . . . . 6 ℤ ∈ V
21pwex 5335 . . . . 5 𝒫 ℤ ∈ V
3 uzf 12766 . . . . . 6 :ℤ⟶𝒫 ℤ
4 frn 6675 . . . . . 6 (ℤ:ℤ⟶𝒫 ℤ → ran ℤ ⊆ 𝒫 ℤ)
53, 4ax-mp 5 . . . . 5 ran ℤ ⊆ 𝒫 ℤ
62, 5ssexi 5279 . . . 4 ran ℤ ∈ V
7 uzfbas.1 . . . . 5 𝑍 = (ℤ𝑀)
87fvexi 6856 . . . 4 𝑍 ∈ V
9 restval 17308 . . . 4 ((ran ℤ ∈ V ∧ 𝑍 ∈ V) → (ran ℤt 𝑍) = ran (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)))
106, 8, 9mp2an 690 . . 3 (ran ℤt 𝑍) = ran (𝑥 ∈ ran ℤ ↦ (𝑥𝑍))
117ineq2i 4169 . . . . . . . . 9 ((ℤ𝑦) ∩ 𝑍) = ((ℤ𝑦) ∩ (ℤ𝑀))
12 uzin 12803 . . . . . . . . . 10 ((𝑦 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((ℤ𝑦) ∩ (ℤ𝑀)) = (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
1312ancoms 459 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑦) ∩ (ℤ𝑀)) = (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
1411, 13eqtrid 2788 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑦) ∩ 𝑍) = (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
15 ffn 6668 . . . . . . . . . 10 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
163, 15ax-mp 5 . . . . . . . . 9 Fn ℤ
17 uzssz 12784 . . . . . . . . . 10 (ℤ𝑀) ⊆ ℤ
187, 17eqsstri 3978 . . . . . . . . 9 𝑍 ⊆ ℤ
19 ifcl 4531 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ ℤ)
20 uzid 12778 . . . . . . . . . . . 12 (if(𝑦𝑀, 𝑀, 𝑦) ∈ ℤ → if(𝑦𝑀, 𝑀, 𝑦) ∈ (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
2119, 20syl 17 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)))
2221, 14eleqtrrd 2841 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ ((ℤ𝑦) ∩ 𝑍))
2322elin2d 4159 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑦𝑀, 𝑀, 𝑦) ∈ 𝑍)
24 fnfvima 7183 . . . . . . . . 9 ((ℤ Fn ℤ ∧ 𝑍 ⊆ ℤ ∧ if(𝑦𝑀, 𝑀, 𝑦) ∈ 𝑍) → (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)) ∈ (ℤ𝑍))
2516, 18, 23, 24mp3an12i 1465 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ‘if(𝑦𝑀, 𝑀, 𝑦)) ∈ (ℤ𝑍))
2614, 25eqeltrd 2838 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍))
2726ralrimiva 3143 . . . . . 6 (𝑀 ∈ ℤ → ∀𝑦 ∈ ℤ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍))
28 ineq1 4165 . . . . . . . . 9 (𝑥 = (ℤ𝑦) → (𝑥𝑍) = ((ℤ𝑦) ∩ 𝑍))
2928eleq1d 2822 . . . . . . . 8 (𝑥 = (ℤ𝑦) → ((𝑥𝑍) ∈ (ℤ𝑍) ↔ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍)))
3029ralrn 7038 . . . . . . 7 (ℤ Fn ℤ → (∀𝑥 ∈ ran ℤ(𝑥𝑍) ∈ (ℤ𝑍) ↔ ∀𝑦 ∈ ℤ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍)))
3116, 30ax-mp 5 . . . . . 6 (∀𝑥 ∈ ran ℤ(𝑥𝑍) ∈ (ℤ𝑍) ↔ ∀𝑦 ∈ ℤ ((ℤ𝑦) ∩ 𝑍) ∈ (ℤ𝑍))
3227, 31sylibr 233 . . . . 5 (𝑀 ∈ ℤ → ∀𝑥 ∈ ran ℤ(𝑥𝑍) ∈ (ℤ𝑍))
33 eqid 2736 . . . . . 6 (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)) = (𝑥 ∈ ran ℤ ↦ (𝑥𝑍))
3433fmpt 7058 . . . . 5 (∀𝑥 ∈ ran ℤ(𝑥𝑍) ∈ (ℤ𝑍) ↔ (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)):ran ℤ⟶(ℤ𝑍))
3532, 34sylib 217 . . . 4 (𝑀 ∈ ℤ → (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)):ran ℤ⟶(ℤ𝑍))
3635frnd 6676 . . 3 (𝑀 ∈ ℤ → ran (𝑥 ∈ ran ℤ ↦ (𝑥𝑍)) ⊆ (ℤ𝑍))
3710, 36eqsstrid 3992 . 2 (𝑀 ∈ ℤ → (ran ℤt 𝑍) ⊆ (ℤ𝑍))
387uztrn2 12782 . . . . . . . . 9 ((𝑥𝑍𝑦 ∈ (ℤ𝑥)) → 𝑦𝑍)
3938ex 413 . . . . . . . 8 (𝑥𝑍 → (𝑦 ∈ (ℤ𝑥) → 𝑦𝑍))
4039ssrdv 3950 . . . . . . 7 (𝑥𝑍 → (ℤ𝑥) ⊆ 𝑍)
4140adantl 482 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ⊆ 𝑍)
42 df-ss 3927 . . . . . 6 ((ℤ𝑥) ⊆ 𝑍 ↔ ((ℤ𝑥) ∩ 𝑍) = (ℤ𝑥))
4341, 42sylib 217 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → ((ℤ𝑥) ∩ 𝑍) = (ℤ𝑥))
4418sseli 3940 . . . . . . . 8 (𝑥𝑍𝑥 ∈ ℤ)
4544adantl 482 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → 𝑥 ∈ ℤ)
46 fnfvelrn 7031 . . . . . . 7 ((ℤ Fn ℤ ∧ 𝑥 ∈ ℤ) → (ℤ𝑥) ∈ ran ℤ)
4716, 45, 46sylancr 587 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ∈ ran ℤ)
48 elrestr 17310 . . . . . 6 ((ran ℤ ∈ V ∧ 𝑍 ∈ V ∧ (ℤ𝑥) ∈ ran ℤ) → ((ℤ𝑥) ∩ 𝑍) ∈ (ran ℤt 𝑍))
496, 8, 47, 48mp3an12i 1465 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → ((ℤ𝑥) ∩ 𝑍) ∈ (ran ℤt 𝑍))
5043, 49eqeltrrd 2839 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑥𝑍) → (ℤ𝑥) ∈ (ran ℤt 𝑍))
5150ralrimiva 3143 . . 3 (𝑀 ∈ ℤ → ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍))
52 ffun 6671 . . . . 5 (ℤ:ℤ⟶𝒫 ℤ → Fun ℤ)
533, 52ax-mp 5 . . . 4 Fun ℤ
543fdmi 6680 . . . . 5 dom ℤ = ℤ
5518, 54sseqtrri 3981 . . . 4 𝑍 ⊆ dom ℤ
56 funimass4 6907 . . . 4 ((Fun ℤ𝑍 ⊆ dom ℤ) → ((ℤ𝑍) ⊆ (ran ℤt 𝑍) ↔ ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍)))
5753, 55, 56mp2an 690 . . 3 ((ℤ𝑍) ⊆ (ran ℤt 𝑍) ↔ ∀𝑥𝑍 (ℤ𝑥) ∈ (ran ℤt 𝑍))
5851, 57sylibr 233 . 2 (𝑀 ∈ ℤ → (ℤ𝑍) ⊆ (ran ℤt 𝑍))
5937, 58eqssd 3961 1 (𝑀 ∈ ℤ → (ran ℤt 𝑍) = (ℤ𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cin 3909  wss 3910  ifcif 4486  𝒫 cpw 4560   class class class wbr 5105  cmpt 5188  dom cdm 5633  ran crn 5634  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cle 11190  cz 12499  cuz 12763  t crest 17302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-pre-lttri 11125  ax-pre-lttrn 11126
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-neg 11388  df-z 12500  df-uz 12764  df-rest 17304
This theorem is referenced by:  uzfbas  23249
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