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Theorem uzrest 20912
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  =  ( ZZ>= `  M )
Assertion
Ref Expression
uzrest  |-  ( M  e.  ZZ  ->  ( ran  ZZ>=t  Z )  =  (
ZZ>= " Z ) )

Proof of Theorem uzrest
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zex 10946 . . . . . 6  |-  ZZ  e.  _V
21pwex 4586 . . . . 5  |-  ~P ZZ  e.  _V
3 uzf 11162 . . . . . 6  |-  ZZ>= : ZZ --> ~P ZZ
4 frn 5735 . . . . . 6  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ran  ZZ>=  C_  ~P ZZ )
53, 4ax-mp 5 . . . . 5  |-  ran  ZZ>=  C_  ~P ZZ
62, 5ssexi 4548 . . . 4  |-  ran  ZZ>=  e. 
_V
7 uzfbas.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
8 fvex 5875 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
97, 8eqeltri 2525 . . . 4  |-  Z  e. 
_V
10 restval 15325 . . . 4  |-  ( ( ran  ZZ>=  e.  _V  /\  Z  e.  _V )  ->  ( ran  ZZ>=t  Z )  =  ran  ( x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) ) )
116, 9, 10mp2an 678 . . 3  |-  ( ran  ZZ>=t  Z )  =  ran  (
x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) )
127ineq2i 3631 . . . . . . . . 9  |-  ( (
ZZ>= `  y )  i^i 
Z )  =  ( ( ZZ>= `  y )  i^i  ( ZZ>= `  M )
)
13 uzin 11191 . . . . . . . . . 10  |-  ( ( y  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( ZZ>= `  y
)  i^i  ( ZZ>= `  M ) )  =  ( ZZ>= `  if (
y  <_  M ,  M ,  y )
) )
1413ancoms 455 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  y
)  i^i  ( ZZ>= `  M ) )  =  ( ZZ>= `  if (
y  <_  M ,  M ,  y )
) )
1512, 14syl5eq 2497 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  y
)  i^i  Z )  =  ( ZZ>= `  if ( y  <_  M ,  M ,  y ) ) )
16 ffn 5728 . . . . . . . . . . 11  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
173, 16ax-mp 5 . . . . . . . . . 10  |-  ZZ>=  Fn  ZZ
1817a1i 11 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  -> 
ZZ>=  Fn  ZZ )
19 uzssz 11178 . . . . . . . . . . 11  |-  ( ZZ>= `  M )  C_  ZZ
207, 19eqsstri 3462 . . . . . . . . . 10  |-  Z  C_  ZZ
2120a1i 11 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  Z  C_  ZZ )
22 inss2 3653 . . . . . . . . . 10  |-  ( (
ZZ>= `  y )  i^i 
Z )  C_  Z
23 ifcl 3923 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  if ( y  <_  M ,  M , 
y )  e.  ZZ )
24 uzid 11173 . . . . . . . . . . . 12  |-  ( if ( y  <_  M ,  M ,  y )  e.  ZZ  ->  if ( y  <_  M ,  M ,  y )  e.  ( ZZ>= `  if ( y  <_  M ,  M ,  y ) ) )
2523, 24syl 17 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  if ( y  <_  M ,  M , 
y )  e.  (
ZZ>= `  if ( y  <_  M ,  M ,  y ) ) )
2625, 15eleqtrrd 2532 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  if ( y  <_  M ,  M , 
y )  e.  ( ( ZZ>= `  y )  i^i  Z ) )
2722, 26sseldi 3430 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  if ( y  <_  M ,  M , 
y )  e.  Z
)
28 fnfvima 6143 . . . . . . . . 9  |-  ( (
ZZ>=  Fn  ZZ  /\  Z  C_  ZZ  /\  if ( y  <_  M ,  M ,  y )  e.  Z )  ->  ( ZZ>=
`  if ( y  <_  M ,  M ,  y ) )  e.  ( ZZ>= " Z
) )
2918, 21, 27, 28syl3anc 1268 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  ( ZZ>= `  if (
y  <_  M ,  M ,  y )
)  e.  ( ZZ>= " Z ) )
3015, 29eqeltrd 2529 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  y
)  i^i  Z )  e.  ( ZZ>= " Z ) )
3130ralrimiva 2802 . . . . . 6  |-  ( M  e.  ZZ  ->  A. y  e.  ZZ  ( ( ZZ>= `  y )  i^i  Z
)  e.  ( ZZ>= " Z ) )
32 ineq1 3627 . . . . . . . . 9  |-  ( x  =  ( ZZ>= `  y
)  ->  ( x  i^i  Z )  =  ( ( ZZ>= `  y )  i^i  Z ) )
3332eleq1d 2513 . . . . . . . 8  |-  ( x  =  ( ZZ>= `  y
)  ->  ( (
x  i^i  Z )  e.  ( ZZ>= " Z )  <->  ( ( ZZ>=
`  y )  i^i 
Z )  e.  (
ZZ>= " Z ) ) )
3433ralrn 6025 . . . . . . 7  |-  ( ZZ>=  Fn  ZZ  ->  ( A. x  e.  ran  ZZ>= ( x  i^i  Z )  e.  ( ZZ>= " Z )  <->  A. y  e.  ZZ  ( ( ZZ>= `  y )  i^i  Z
)  e.  ( ZZ>= " Z ) ) )
3517, 34ax-mp 5 . . . . . 6  |-  ( A. x  e.  ran  ZZ>= ( x  i^i  Z )  e.  ( ZZ>= " Z )  <->  A. y  e.  ZZ  ( ( ZZ>= `  y )  i^i  Z
)  e.  ( ZZ>= " Z ) )
3631, 35sylibr 216 . . . . 5  |-  ( M  e.  ZZ  ->  A. x  e.  ran  ZZ>= ( x  i^i 
Z )  e.  (
ZZ>= " Z ) )
37 eqid 2451 . . . . . 6  |-  ( x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) )  =  ( x  e. 
ran  ZZ>=  |->  ( x  i^i 
Z ) )
3837fmpt 6043 . . . . 5  |-  ( A. x  e.  ran  ZZ>= ( x  i^i  Z )  e.  ( ZZ>= " Z )  <->  ( x  e.  ran  ZZ>=  |->  ( x  i^i 
Z ) ) : ran  ZZ>= --> ( ZZ>= " Z
) )
3936, 38sylib 200 . . . 4  |-  ( M  e.  ZZ  ->  (
x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) ) : ran  ZZ>= --> (
ZZ>= " Z ) )
40 frn 5735 . . . 4  |-  ( ( x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) ) : ran  ZZ>= --> (
ZZ>= " Z )  ->  ran  ( x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) )  C_  ( ZZ>= " Z ) )
4139, 40syl 17 . . 3  |-  ( M  e.  ZZ  ->  ran  ( x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) )  C_  ( ZZ>= " Z ) )
4211, 41syl5eqss 3476 . 2  |-  ( M  e.  ZZ  ->  ( ran  ZZ>=t  Z )  C_  ( ZZ>=
" Z ) )
437uztrn2 11176 . . . . . . . . 9  |-  ( ( x  e.  Z  /\  y  e.  ( ZZ>= `  x ) )  -> 
y  e.  Z )
4443ex 436 . . . . . . . 8  |-  ( x  e.  Z  ->  (
y  e.  ( ZZ>= `  x )  ->  y  e.  Z ) )
4544ssrdv 3438 . . . . . . 7  |-  ( x  e.  Z  ->  ( ZZ>=
`  x )  C_  Z )
4645adantl 468 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ZZ>= `  x )  C_  Z )
47 df-ss 3418 . . . . . 6  |-  ( (
ZZ>= `  x )  C_  Z 
<->  ( ( ZZ>= `  x
)  i^i  Z )  =  ( ZZ>= `  x
) )
4846, 47sylib 200 . . . . 5  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ( ZZ>= `  x
)  i^i  Z )  =  ( ZZ>= `  x
) )
496a1i 11 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ran  ZZ>=  e.  _V )
509a1i 11 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  Z  e.  _V )
5120sseli 3428 . . . . . . . 8  |-  ( x  e.  Z  ->  x  e.  ZZ )
5251adantl 468 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  x  e.  ZZ )
53 fnfvelrn 6019 . . . . . . 7  |-  ( (
ZZ>=  Fn  ZZ  /\  x  e.  ZZ )  ->  ( ZZ>=
`  x )  e. 
ran  ZZ>= )
5417, 52, 53sylancr 669 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ZZ>= `  x )  e.  ran  ZZ>= )
55 elrestr 15327 . . . . . 6  |-  ( ( ran  ZZ>=  e.  _V  /\  Z  e.  _V  /\  ( ZZ>= `  x )  e.  ran  ZZ>= )  ->  ( ( ZZ>= `  x )  i^i  Z
)  e.  ( ran  ZZ>=t  Z ) )
5649, 50, 54, 55syl3anc 1268 . . . . 5  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ( ZZ>= `  x
)  i^i  Z )  e.  ( ran  ZZ>=t  Z ) )
5748, 56eqeltrrd 2530 . . . 4  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ZZ>= `  x )  e.  ( ran  ZZ>=t  Z ) )
5857ralrimiva 2802 . . 3  |-  ( M  e.  ZZ  ->  A. x  e.  Z  ( ZZ>= `  x )  e.  ( ran  ZZ>=t  Z ) )
59 ffun 5731 . . . . 5  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  Fun  ZZ>= )
603, 59ax-mp 5 . . . 4  |-  Fun  ZZ>=
613fdmi 5734 . . . . 5  |-  dom  ZZ>=  =  ZZ
6220, 61sseqtr4i 3465 . . . 4  |-  Z  C_  dom  ZZ>=
63 funimass4 5916 . . . 4  |-  ( ( Fun  ZZ>=  /\  Z  C_  dom  ZZ>= )  ->  ( ( ZZ>= " Z )  C_  ( ran  ZZ>=t  Z )  <->  A. x  e.  Z  ( ZZ>= `  x )  e.  ( ran  ZZ>=t  Z ) ) )
6460, 62, 63mp2an 678 . . 3  |-  ( (
ZZ>= " Z )  C_  ( ran  ZZ>=t  Z )  <->  A. x  e.  Z  ( ZZ>= `  x )  e.  ( ran  ZZ>=t  Z ) )
6558, 64sylibr 216 . 2  |-  ( M  e.  ZZ  ->  ( ZZ>=
" Z )  C_  ( ran  ZZ>=t  Z ) )
6642, 65eqssd 3449 1  |-  ( M  e.  ZZ  ->  ( ran  ZZ>=t  Z )  =  (
ZZ>= " Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   _Vcvv 3045    i^i cin 3403    C_ wss 3404   ifcif 3881   ~Pcpw 3951   class class class wbr 4402    |-> cmpt 4461   dom cdm 4834   ran crn 4835   "cima 4837   Fun wfun 5576    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290    <_ cle 9676   ZZcz 10937   ZZ>=cuz 11159   ↾t crest 15319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-pre-lttri 9613  ax-pre-lttrn 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-neg 9863  df-z 10938  df-uz 11160  df-rest 15321
This theorem is referenced by:  uzfbas  20913
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