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Theorem uzrest 20524
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 10894 . . . . . 6  |-  ZZ  e.  _V
21pwex 4639 . . . . 5  |-  ~P ZZ  e.  _V
3 uzf 11109 . . . . . 6  |-  ZZ>= : ZZ --> ~P ZZ
4 frn 5743 . . . . . 6  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ran  ZZ>=  C_  ~P ZZ )
53, 4ax-mp 5 . . . . 5  |-  ran  ZZ>=  C_  ~P ZZ
62, 5ssexi 4601 . . . 4  |-  ran  ZZ>=  e. 
_V
7 uzfbas.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
8 fvex 5882 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
97, 8eqeltri 2541 . . . 4  |-  Z  e. 
_V
10 restval 14844 . . . 4  |-  ( ( ran  ZZ>=  e.  _V  /\  Z  e.  _V )  ->  ( ran  ZZ>=t  Z )  =  ran  ( x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) ) )
116, 9, 10mp2an 672 . . 3  |-  ( ran  ZZ>=t  Z )  =  ran  (
x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) )
127ineq2i 3693 . . . . . . . . 9  |-  ( (
ZZ>= `  y )  i^i 
Z )  =  ( ( ZZ>= `  y )  i^i  ( ZZ>= `  M )
)
13 uzin 11138 . . . . . . . . . 10  |-  ( ( y  e.  ZZ  /\  M  e.  ZZ )  ->  ( ( ZZ>= `  y
)  i^i  ( ZZ>= `  M ) )  =  ( ZZ>= `  if (
y  <_  M ,  M ,  y )
) )
1413ancoms 453 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  y
)  i^i  ( ZZ>= `  M ) )  =  ( ZZ>= `  if (
y  <_  M ,  M ,  y )
) )
1512, 14syl5eq 2510 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  y
)  i^i  Z )  =  ( ZZ>= `  if ( y  <_  M ,  M ,  y ) ) )
16 ffn 5737 . . . . . . . . . . 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 11125 . . . . . . . . . . 11  |-  ( ZZ>= `  M )  C_  ZZ
207, 19eqsstri 3529 . . . . . . . . . 10  |-  Z  C_  ZZ
2120a1i 11 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  Z  C_  ZZ )
22 inss2 3715 . . . . . . . . . 10  |-  ( (
ZZ>= `  y )  i^i 
Z )  C_  Z
23 ifcl 3986 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  if ( y  <_  M ,  M , 
y )  e.  ZZ )
24 uzid 11120 . . . . . . . . . . . 12  |-  ( if ( y  <_  M ,  M ,  y )  e.  ZZ  ->  if ( y  <_  M ,  M ,  y )  e.  ( ZZ>= `  if ( y  <_  M ,  M ,  y ) ) )
2523, 24syl 16 . . . . . . . . . . 11  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  if ( y  <_  M ,  M , 
y )  e.  (
ZZ>= `  if ( y  <_  M ,  M ,  y ) ) )
2625, 15eleqtrrd 2548 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  if ( y  <_  M ,  M , 
y )  e.  ( ( ZZ>= `  y )  i^i  Z ) )
2722, 26sseldi 3497 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  if ( y  <_  M ,  M , 
y )  e.  Z
)
28 fnfvima 6151 . . . . . . . . 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 1228 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  ( ZZ>= `  if (
y  <_  M ,  M ,  y )
)  e.  ( ZZ>= " Z ) )
3015, 29eqeltrd 2545 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  y
)  i^i  Z )  e.  ( ZZ>= " Z ) )
3130ralrimiva 2871 . . . . . 6  |-  ( M  e.  ZZ  ->  A. y  e.  ZZ  ( ( ZZ>= `  y )  i^i  Z
)  e.  ( ZZ>= " Z ) )
32 ineq1 3689 . . . . . . . . 9  |-  ( x  =  ( ZZ>= `  y
)  ->  ( x  i^i  Z )  =  ( ( ZZ>= `  y )  i^i  Z ) )
3332eleq1d 2526 . . . . . . . 8  |-  ( x  =  ( ZZ>= `  y
)  ->  ( (
x  i^i  Z )  e.  ( ZZ>= " Z )  <->  ( ( ZZ>=
`  y )  i^i 
Z )  e.  (
ZZ>= " Z ) ) )
3433ralrn 6035 . . . . . . 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 212 . . . . 5  |-  ( M  e.  ZZ  ->  A. x  e.  ran  ZZ>= ( x  i^i 
Z )  e.  (
ZZ>= " Z ) )
37 eqid 2457 . . . . . 6  |-  ( x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) )  =  ( x  e. 
ran  ZZ>=  |->  ( x  i^i 
Z ) )
3837fmpt 6053 . . . . 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 196 . . . 4  |-  ( M  e.  ZZ  ->  (
x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) ) : ran  ZZ>= --> (
ZZ>= " Z ) )
40 frn 5743 . . . 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 16 . . 3  |-  ( M  e.  ZZ  ->  ran  ( x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) )  C_  ( ZZ>= " Z ) )
4211, 41syl5eqss 3543 . 2  |-  ( M  e.  ZZ  ->  ( ran  ZZ>=t  Z )  C_  ( ZZ>=
" Z ) )
437uztrn2 11123 . . . . . . . . 9  |-  ( ( x  e.  Z  /\  y  e.  ( ZZ>= `  x ) )  -> 
y  e.  Z )
4443ex 434 . . . . . . . 8  |-  ( x  e.  Z  ->  (
y  e.  ( ZZ>= `  x )  ->  y  e.  Z ) )
4544ssrdv 3505 . . . . . . 7  |-  ( x  e.  Z  ->  ( ZZ>=
`  x )  C_  Z )
4645adantl 466 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ZZ>= `  x )  C_  Z )
47 df-ss 3485 . . . . . 6  |-  ( (
ZZ>= `  x )  C_  Z 
<->  ( ( ZZ>= `  x
)  i^i  Z )  =  ( ZZ>= `  x
) )
4846, 47sylib 196 . . . . 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 3495 . . . . . . . 8  |-  ( x  e.  Z  ->  x  e.  ZZ )
5251adantl 466 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  x  e.  ZZ )
53 fnfvelrn 6029 . . . . . . 7  |-  ( (
ZZ>=  Fn  ZZ  /\  x  e.  ZZ )  ->  ( ZZ>=
`  x )  e. 
ran  ZZ>= )
5417, 52, 53sylancr 663 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ZZ>= `  x )  e.  ran  ZZ>= )
55 elrestr 14846 . . . . . 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 1228 . . . . 5  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ( ZZ>= `  x
)  i^i  Z )  e.  ( ran  ZZ>=t  Z ) )
5748, 56eqeltrrd 2546 . . . 4  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ZZ>= `  x )  e.  ( ran  ZZ>=t  Z ) )
5857ralrimiva 2871 . . 3  |-  ( M  e.  ZZ  ->  A. x  e.  Z  ( ZZ>= `  x )  e.  ( ran  ZZ>=t  Z ) )
59 ffun 5739 . . . . 5  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  Fun  ZZ>= )
603, 59ax-mp 5 . . . 4  |-  Fun  ZZ>=
613fdmi 5742 . . . . 5  |-  dom  ZZ>=  =  ZZ
6220, 61sseqtr4i 3532 . . . 4  |-  Z  C_  dom  ZZ>=
63 funimass4 5924 . . . 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 672 . . 3  |-  ( (
ZZ>= " Z )  C_  ( ran  ZZ>=t  Z )  <->  A. x  e.  Z  ( ZZ>= `  x )  e.  ( ran  ZZ>=t  Z ) )
6558, 64sylibr 212 . 2  |-  ( M  e.  ZZ  ->  ( ZZ>=
" Z )  C_  ( ran  ZZ>=t  Z ) )
6642, 65eqssd 3516 1  |-  ( M  e.  ZZ  ->  ( ran  ZZ>=t  Z )  =  (
ZZ>= " Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    i^i cin 3470    C_ wss 3471   ifcif 3944   ~Pcpw 4015   class class class wbr 4456    |-> cmpt 4515   dom cdm 5008   ran crn 5009   "cima 5011   Fun wfun 5588    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    <_ cle 9646   ZZcz 10885   ZZ>=cuz 11106   ↾t crest 14838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-neg 9827  df-z 10886  df-uz 11107  df-rest 14840
This theorem is referenced by:  uzfbas  20525
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