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Theorem uzrest 19612
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 10770 . . . . . 6  |-  ZZ  e.  _V
21pwex 4586 . . . . 5  |-  ~P ZZ  e.  _V
3 uzf 10979 . . . . . 6  |-  ZZ>= : ZZ --> ~P ZZ
4 frn 5676 . . . . . 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 5812 . . . . 5  |-  ( ZZ>= `  M )  e.  _V
97, 8eqeltri 2538 . . . 4  |-  Z  e. 
_V
10 restval 14488 . . . 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 3660 . . . . . . . . 9  |-  ( (
ZZ>= `  y )  i^i 
Z )  =  ( ( ZZ>= `  y )  i^i  ( ZZ>= `  M )
)
13 uzin 11008 . . . . . . . . . 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 2507 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  y
)  i^i  Z )  =  ( ZZ>= `  if ( y  <_  M ,  M ,  y ) ) )
16 ffn 5670 . . . . . . . . . . 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 10995 . . . . . . . . . . 11  |-  ( ZZ>= `  M )  C_  ZZ
207, 19eqsstri 3497 . . . . . . . . . 10  |-  Z  C_  ZZ
2120a1i 11 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  Z  C_  ZZ )
22 inss2 3682 . . . . . . . . . 10  |-  ( (
ZZ>= `  y )  i^i 
Z )  C_  Z
23 ifcl 3942 . . . . . . . . . . . 12  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  if ( y  <_  M ,  M , 
y )  e.  ZZ )
24 uzid 10990 . . . . . . . . . . . 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 2545 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  if ( y  <_  M ,  M , 
y )  e.  ( ( ZZ>= `  y )  i^i  Z ) )
2722, 26sseldi 3465 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  if ( y  <_  M ,  M , 
y )  e.  Z
)
28 fnfvima 6067 . . . . . . . . 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 1219 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  ( ZZ>= `  if (
y  <_  M ,  M ,  y )
)  e.  ( ZZ>= " Z ) )
3015, 29eqeltrd 2542 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  y
)  i^i  Z )  e.  ( ZZ>= " Z ) )
3130ralrimiva 2830 . . . . . 6  |-  ( M  e.  ZZ  ->  A. y  e.  ZZ  ( ( ZZ>= `  y )  i^i  Z
)  e.  ( ZZ>= " Z ) )
32 ineq1 3656 . . . . . . . . 9  |-  ( x  =  ( ZZ>= `  y
)  ->  ( x  i^i  Z )  =  ( ( ZZ>= `  y )  i^i  Z ) )
3332eleq1d 2523 . . . . . . . 8  |-  ( x  =  ( ZZ>= `  y
)  ->  ( (
x  i^i  Z )  e.  ( ZZ>= " Z )  <->  ( ( ZZ>=
`  y )  i^i 
Z )  e.  (
ZZ>= " Z ) ) )
3433ralrn 5958 . . . . . . 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 2454 . . . . . 6  |-  ( x  e.  ran  ZZ>=  |->  ( x  i^i  Z ) )  =  ( x  e. 
ran  ZZ>=  |->  ( x  i^i 
Z ) )
3837fmpt 5976 . . . . 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 5676 . . . 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 3511 . 2  |-  ( M  e.  ZZ  ->  ( ran  ZZ>=t  Z )  C_  ( ZZ>=
" Z ) )
437uztrn2 10993 . . . . . . . . 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 3473 . . . . . . 7  |-  ( x  e.  Z  ->  ( ZZ>=
`  x )  C_  Z )
4645adantl 466 . . . . . 6  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ZZ>= `  x )  C_  Z )
47 df-ss 3453 . . . . . 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 3463 . . . . . . . 8  |-  ( x  e.  Z  ->  x  e.  ZZ )
5251adantl 466 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  x  e.  ZZ )
53 fnfvelrn 5952 . . . . . . 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 14490 . . . . . 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 1219 . . . . 5  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ( ZZ>= `  x
)  i^i  Z )  e.  ( ran  ZZ>=t  Z ) )
5748, 56eqeltrrd 2543 . . . 4  |-  ( ( M  e.  ZZ  /\  x  e.  Z )  ->  ( ZZ>= `  x )  e.  ( ran  ZZ>=t  Z ) )
5857ralrimiva 2830 . . 3  |-  ( M  e.  ZZ  ->  A. x  e.  Z  ( ZZ>= `  x )  e.  ( ran  ZZ>=t  Z ) )
59 ffun 5672 . . . . 5  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  Fun  ZZ>= )
603, 59ax-mp 5 . . . 4  |-  Fun  ZZ>=
613fdmi 5675 . . . . 5  |-  dom  ZZ>=  =  ZZ
6220, 61sseqtr4i 3500 . . . 4  |-  Z  C_  dom  ZZ>=
63 funimass4 5854 . . . 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 3484 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 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078    i^i cin 3438    C_ wss 3439   ifcif 3902   ~Pcpw 3971   class class class wbr 4403    |-> cmpt 4461   dom cdm 4951   ran crn 4952   "cima 4954   Fun wfun 5523    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    <_ cle 9534   ZZcz 10761   ZZ>=cuz 10976   ↾t crest 14482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-pre-lttri 9471  ax-pre-lttrn 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-neg 9713  df-z 10762  df-uz 10977  df-rest 14484
This theorem is referenced by:  uzfbas  19613
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