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Theorem uzrest 23271
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 12516 . . . . . 6 β„€ ∈ V
21pwex 5339 . . . . 5 𝒫 β„€ ∈ V
3 uzf 12774 . . . . . 6 β„€β‰₯:β„€βŸΆπ’« β„€
4 frn 6679 . . . . . 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 5283 . . . 4 ran β„€β‰₯ ∈ V
7 uzfbas.1 . . . . 5 𝑍 = (β„€β‰₯β€˜π‘€)
87fvexi 6860 . . . 4 𝑍 ∈ V
9 restval 17316 . . . 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 4173 . . . . . . . . 9 ((β„€β‰₯β€˜π‘¦) ∩ 𝑍) = ((β„€β‰₯β€˜π‘¦) ∩ (β„€β‰₯β€˜π‘€))
12 uzin 12811 . . . . . . . . . 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 6672 . . . . . . . . . 10 (β„€β‰₯:β„€βŸΆπ’« β„€ β†’ β„€β‰₯ Fn β„€)
163, 15ax-mp 5 . . . . . . . . 9 β„€β‰₯ Fn β„€
17 uzssz 12792 . . . . . . . . . 10 (β„€β‰₯β€˜π‘€) βŠ† β„€
187, 17eqsstri 3982 . . . . . . . . 9 𝑍 βŠ† β„€
19 ifcl 4535 . . . . . . . . . . . 12 ((𝑀 ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ if(𝑦 ≀ 𝑀, 𝑀, 𝑦) ∈ β„€)
20 uzid 12786 . . . . . . . . . . . 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 4163 . . . . . . . . 9 ((𝑀 ∈ β„€ ∧ 𝑦 ∈ β„€) β†’ if(𝑦 ≀ 𝑀, 𝑀, 𝑦) ∈ 𝑍)
24 fnfvima 7187 . . . . . . . . 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 3140 . . . . . 6 (𝑀 ∈ β„€ β†’ βˆ€π‘¦ ∈ β„€ ((β„€β‰₯β€˜π‘¦) ∩ 𝑍) ∈ (β„€β‰₯ β€œ 𝑍))
28 ineq1 4169 . . . . . . . . 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 7042 . . . . . . 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 7062 . . . . 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 6680 . . 3 (𝑀 ∈ β„€ β†’ ran (π‘₯ ∈ ran β„€β‰₯ ↦ (π‘₯ ∩ 𝑍)) βŠ† (β„€β‰₯ β€œ 𝑍))
3710, 36eqsstrid 3996 . 2 (𝑀 ∈ β„€ β†’ (ran β„€β‰₯ β†Ύt 𝑍) βŠ† (β„€β‰₯ β€œ 𝑍))
387uztrn2 12790 . . . . . . . . 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 3954 . . . . . . 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 3931 . . . . . 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 3944 . . . . . . . 8 (π‘₯ ∈ 𝑍 β†’ π‘₯ ∈ β„€)
4544adantl 483 . . . . . . 7 ((𝑀 ∈ β„€ ∧ π‘₯ ∈ 𝑍) β†’ π‘₯ ∈ β„€)
46 fnfvelrn 7035 . . . . . . 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 17318 . . . . . 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 3140 . . 3 (𝑀 ∈ β„€ β†’ βˆ€π‘₯ ∈ 𝑍 (β„€β‰₯β€˜π‘₯) ∈ (ran β„€β‰₯ β†Ύt 𝑍))
52 ffun 6675 . . . . 5 (β„€β‰₯:β„€βŸΆπ’« β„€ β†’ Fun β„€β‰₯)
533, 52ax-mp 5 . . . 4 Fun β„€β‰₯
543fdmi 6684 . . . . 5 dom β„€β‰₯ = β„€
5518, 54sseqtrri 3985 . . . 4 𝑍 βŠ† dom β„€β‰₯
56 funimass4 6911 . . . 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 3965 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 3061  Vcvv 3447   ∩ cin 3913   βŠ† wss 3914  ifcif 4490  π’« cpw 4564   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5637  ran crn 5638   β€œ cima 5640  Fun wfun 6494   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ≀ cle 11198  β„€cz 12507  β„€β‰₯cuz 12771   β†Ύt crest 17310
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-pre-lttri 11133  ax-pre-lttrn 11134
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-po 5549  df-so 5550  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-neg 11396  df-z 12508  df-uz 12772  df-rest 17312
This theorem is referenced by:  uzfbas  23272
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