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Theorem uzrest 23401
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 = (β„€β‰₯β€˜π‘€)
Assertion
Ref Expression
uzrest (𝑀 ∈ β„€ β†’ (ran β„€β‰₯ β†Ύt 𝑍) = (β„€β‰₯ β€œ 𝑍))

Proof of Theorem uzrest
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zex 12567 . . . . . 6 β„€ ∈ V
21pwex 5379 . . . . 5 𝒫 β„€ ∈ V
3 uzf 12825 . . . . . 6 β„€β‰₯:β„€βŸΆπ’« β„€
4 frn 6725 . . . . . 6 (β„€β‰₯:β„€βŸΆπ’« β„€ β†’ ran β„€β‰₯ βŠ† 𝒫 β„€)
53, 4ax-mp 5 . . . . 5 ran β„€β‰₯ βŠ† 𝒫 β„€
62, 5ssexi 5323 . . . 4 ran β„€β‰₯ ∈ V
7 uzfbas.1 . . . . 5 𝑍 = (β„€β‰₯β€˜π‘€)
87fvexi 6906 . . . 4 𝑍 ∈ V
9 restval 17372 . . . 4 ((ran β„€β‰₯ ∈ V ∧ 𝑍 ∈ V) β†’ (ran β„€β‰₯ β†Ύt 𝑍) = ran (π‘₯ ∈ ran β„€β‰₯ ↦ (π‘₯ ∩ 𝑍)))
106, 8, 9mp2an 691 . . 3 (ran β„€β‰₯ β†Ύt 𝑍) = ran (π‘₯ ∈ ran β„€β‰₯ ↦ (π‘₯ ∩ 𝑍))
117ineq2i 4210 . . . . . . . . 9 ((β„€β‰₯β€˜π‘¦) ∩ 𝑍) = ((β„€β‰₯β€˜π‘¦) ∩ (β„€β‰₯β€˜π‘€))
12 uzin 12862 . . . . . . . . . 10 ((𝑦 ∈ β„€ ∧ 𝑀 ∈ β„€) β†’ ((β„€β‰₯β€˜π‘¦) ∩ (β„€β‰₯β€˜π‘€)) = (β„€β‰₯β€˜if(𝑦 ≀ 𝑀, 𝑀, 𝑦)))
1312ancoms 460 . . . . . . . . 9 ((𝑀 ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ ((β„€β‰₯β€˜π‘¦) ∩ (β„€β‰₯β€˜π‘€)) = (β„€β‰₯β€˜if(𝑦 ≀ 𝑀, 𝑀, 𝑦)))
1411, 13eqtrid 2785 . . . . . . . 8 ((𝑀 ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ ((β„€β‰₯β€˜π‘¦) ∩ 𝑍) = (β„€β‰₯β€˜if(𝑦 ≀ 𝑀, 𝑀, 𝑦)))
15 ffn 6718 . . . . . . . . . 10 (β„€β‰₯:β„€βŸΆπ’« β„€ β†’ β„€β‰₯ Fn β„€)
163, 15ax-mp 5 . . . . . . . . 9 β„€β‰₯ Fn β„€
17 uzssz 12843 . . . . . . . . . 10 (β„€β‰₯β€˜π‘€) βŠ† β„€
187, 17eqsstri 4017 . . . . . . . . 9 𝑍 βŠ† β„€
19 ifcl 4574 . . . . . . . . . . . 12 ((𝑀 ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ if(𝑦 ≀ 𝑀, 𝑀, 𝑦) ∈ β„€)
20 uzid 12837 . . . . . . . . . . . 12 (if(𝑦 ≀ 𝑀, 𝑀, 𝑦) ∈ β„€ β†’ if(𝑦 ≀ 𝑀, 𝑀, 𝑦) ∈ (β„€β‰₯β€˜if(𝑦 ≀ 𝑀, 𝑀, 𝑦)))
2119, 20syl 17 . . . . . . . . . . 11 ((𝑀 ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ if(𝑦 ≀ 𝑀, 𝑀, 𝑦) ∈ (β„€β‰₯β€˜if(𝑦 ≀ 𝑀, 𝑀, 𝑦)))
2221, 14eleqtrrd 2837 . . . . . . . . . 10 ((𝑀 ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ if(𝑦 ≀ 𝑀, 𝑀, 𝑦) ∈ ((β„€β‰₯β€˜π‘¦) ∩ 𝑍))
2322elin2d 4200 . . . . . . . . 9 ((𝑀 ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ if(𝑦 ≀ 𝑀, 𝑀, 𝑦) ∈ 𝑍)
24 fnfvima 7235 . . . . . . . . 9 ((β„€β‰₯ Fn β„€ ∧ 𝑍 βŠ† β„€ ∧ if(𝑦 ≀ 𝑀, 𝑀, 𝑦) ∈ 𝑍) β†’ (β„€β‰₯β€˜if(𝑦 ≀ 𝑀, 𝑀, 𝑦)) ∈ (β„€β‰₯ β€œ 𝑍))
2516, 18, 23, 24mp3an12i 1466 . . . . . . . 8 ((𝑀 ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ (β„€β‰₯β€˜if(𝑦 ≀ 𝑀, 𝑀, 𝑦)) ∈ (β„€β‰₯ β€œ 𝑍))
2614, 25eqeltrd 2834 . . . . . . 7 ((𝑀 ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ ((β„€β‰₯β€˜π‘¦) ∩ 𝑍) ∈ (β„€β‰₯ β€œ 𝑍))
2726ralrimiva 3147 . . . . . 6 (𝑀 ∈ β„€ β†’ βˆ€π‘¦ ∈ β„€ ((β„€β‰₯β€˜π‘¦) ∩ 𝑍) ∈ (β„€β‰₯ β€œ 𝑍))
28 ineq1 4206 . . . . . . . . 9 (π‘₯ = (β„€β‰₯β€˜π‘¦) β†’ (π‘₯ ∩ 𝑍) = ((β„€β‰₯β€˜π‘¦) ∩ 𝑍))
2928eleq1d 2819 . . . . . . . 8 (π‘₯ = (β„€β‰₯β€˜π‘¦) β†’ ((π‘₯ ∩ 𝑍) ∈ (β„€β‰₯ β€œ 𝑍) ↔ ((β„€β‰₯β€˜π‘¦) ∩ 𝑍) ∈ (β„€β‰₯ β€œ 𝑍)))
3029ralrn 7090 . . . . . . 7 (β„€β‰₯ Fn β„€ β†’ (βˆ€π‘₯ ∈ ran β„€β‰₯(π‘₯ ∩ 𝑍) ∈ (β„€β‰₯ β€œ 𝑍) ↔ βˆ€π‘¦ ∈ β„€ ((β„€β‰₯β€˜π‘¦) ∩ 𝑍) ∈ (β„€β‰₯ β€œ 𝑍)))
3116, 30ax-mp 5 . . . . . 6 (βˆ€π‘₯ ∈ ran β„€β‰₯(π‘₯ ∩ 𝑍) ∈ (β„€β‰₯ β€œ 𝑍) ↔ βˆ€π‘¦ ∈ β„€ ((β„€β‰₯β€˜π‘¦) ∩ 𝑍) ∈ (β„€β‰₯ β€œ 𝑍))
3227, 31sylibr 233 . . . . 5 (𝑀 ∈ β„€ β†’ βˆ€π‘₯ ∈ ran β„€β‰₯(π‘₯ ∩ 𝑍) ∈ (β„€β‰₯ β€œ 𝑍))
33 eqid 2733 . . . . . 6 (π‘₯ ∈ ran β„€β‰₯ ↦ (π‘₯ ∩ 𝑍)) = (π‘₯ ∈ ran β„€β‰₯ ↦ (π‘₯ ∩ 𝑍))
3433fmpt 7110 . . . . 5 (βˆ€π‘₯ ∈ ran β„€β‰₯(π‘₯ ∩ 𝑍) ∈ (β„€β‰₯ β€œ 𝑍) ↔ (π‘₯ ∈ ran β„€β‰₯ ↦ (π‘₯ ∩ 𝑍)):ran β„€β‰₯⟢(β„€β‰₯ β€œ 𝑍))
3532, 34sylib 217 . . . 4 (𝑀 ∈ β„€ β†’ (π‘₯ ∈ ran β„€β‰₯ ↦ (π‘₯ ∩ 𝑍)):ran β„€β‰₯⟢(β„€β‰₯ β€œ 𝑍))
3635frnd 6726 . . 3 (𝑀 ∈ β„€ β†’ ran (π‘₯ ∈ ran β„€β‰₯ ↦ (π‘₯ ∩ 𝑍)) βŠ† (β„€β‰₯ β€œ 𝑍))
3710, 36eqsstrid 4031 . 2 (𝑀 ∈ β„€ β†’ (ran β„€β‰₯ β†Ύt 𝑍) βŠ† (β„€β‰₯ β€œ 𝑍))
387uztrn2 12841 . . . . . . . . 9 ((π‘₯ ∈ 𝑍 ∧ 𝑦 ∈ (β„€β‰₯β€˜π‘₯)) β†’ 𝑦 ∈ 𝑍)
3938ex 414 . . . . . . . 8 (π‘₯ ∈ 𝑍 β†’ (𝑦 ∈ (β„€β‰₯β€˜π‘₯) β†’ 𝑦 ∈ 𝑍))
4039ssrdv 3989 . . . . . . 7 (π‘₯ ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘₯) βŠ† 𝑍)
4140adantl 483 . . . . . 6 ((𝑀 ∈ β„€ ∧ π‘₯ ∈ 𝑍) β†’ (β„€β‰₯β€˜π‘₯) βŠ† 𝑍)
42 df-ss 3966 . . . . . 6 ((β„€β‰₯β€˜π‘₯) βŠ† 𝑍 ↔ ((β„€β‰₯β€˜π‘₯) ∩ 𝑍) = (β„€β‰₯β€˜π‘₯))
4341, 42sylib 217 . . . . 5 ((𝑀 ∈ β„€ ∧ π‘₯ ∈ 𝑍) β†’ ((β„€β‰₯β€˜π‘₯) ∩ 𝑍) = (β„€β‰₯β€˜π‘₯))
4418sseli 3979 . . . . . . . 8 (π‘₯ ∈ 𝑍 β†’ π‘₯ ∈ β„€)
4544adantl 483 . . . . . . 7 ((𝑀 ∈ β„€ ∧ π‘₯ ∈ 𝑍) β†’ π‘₯ ∈ β„€)
46 fnfvelrn 7083 . . . . . . 7 ((β„€β‰₯ Fn β„€ ∧ π‘₯ ∈ β„€) β†’ (β„€β‰₯β€˜π‘₯) ∈ ran β„€β‰₯)
4716, 45, 46sylancr 588 . . . . . 6 ((𝑀 ∈ β„€ ∧ π‘₯ ∈ 𝑍) β†’ (β„€β‰₯β€˜π‘₯) ∈ ran β„€β‰₯)
48 elrestr 17374 . . . . . 6 ((ran β„€β‰₯ ∈ V ∧ 𝑍 ∈ V ∧ (β„€β‰₯β€˜π‘₯) ∈ ran β„€β‰₯) β†’ ((β„€β‰₯β€˜π‘₯) ∩ 𝑍) ∈ (ran β„€β‰₯ β†Ύt 𝑍))
496, 8, 47, 48mp3an12i 1466 . . . . 5 ((𝑀 ∈ β„€ ∧ π‘₯ ∈ 𝑍) β†’ ((β„€β‰₯β€˜π‘₯) ∩ 𝑍) ∈ (ran β„€β‰₯ β†Ύt 𝑍))
5043, 49eqeltrrd 2835 . . . 4 ((𝑀 ∈ β„€ ∧ π‘₯ ∈ 𝑍) β†’ (β„€β‰₯β€˜π‘₯) ∈ (ran β„€β‰₯ β†Ύt 𝑍))
5150ralrimiva 3147 . . 3 (𝑀 ∈ β„€ β†’ βˆ€π‘₯ ∈ 𝑍 (β„€β‰₯β€˜π‘₯) ∈ (ran β„€β‰₯ β†Ύt 𝑍))
52 ffun 6721 . . . . 5 (β„€β‰₯:β„€βŸΆπ’« β„€ β†’ Fun β„€β‰₯)
533, 52ax-mp 5 . . . 4 Fun β„€β‰₯
543fdmi 6730 . . . . 5 dom β„€β‰₯ = β„€
5518, 54sseqtrri 4020 . . . 4 𝑍 βŠ† dom β„€β‰₯
56 funimass4 6957 . . . 4 ((Fun β„€β‰₯ ∧ 𝑍 βŠ† dom β„€β‰₯) β†’ ((β„€β‰₯ β€œ 𝑍) βŠ† (ran β„€β‰₯ β†Ύt 𝑍) ↔ βˆ€π‘₯ ∈ 𝑍 (β„€β‰₯β€˜π‘₯) ∈ (ran β„€β‰₯ β†Ύt 𝑍)))
5753, 55, 56mp2an 691 . . 3 ((β„€β‰₯ β€œ 𝑍) βŠ† (ran β„€β‰₯ β†Ύt 𝑍) ↔ βˆ€π‘₯ ∈ 𝑍 (β„€β‰₯β€˜π‘₯) ∈ (ran β„€β‰₯ β†Ύt 𝑍))
5851, 57sylibr 233 . 2 (𝑀 ∈ β„€ β†’ (β„€β‰₯ β€œ 𝑍) βŠ† (ran β„€β‰₯ β†Ύt 𝑍))
5937, 58eqssd 4000 1 (𝑀 ∈ β„€ β†’ (ran β„€β‰₯ β†Ύt 𝑍) = (β„€β‰₯ β€œ 𝑍))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   ∩ cin 3948   βŠ† wss 3949  ifcif 4529  π’« cpw 4603   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677  ran crn 5678   β€œ cima 5680  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ≀ cle 11249  β„€cz 12558  β„€β‰₯cuz 12822   β†Ύt crest 17366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-pre-lttri 11184  ax-pre-lttrn 11185
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-neg 11447  df-z 12559  df-uz 12823  df-rest 17368
This theorem is referenced by:  uzfbas  23402
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