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Theorem uzin2 10474
Description: The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
Assertion
Ref Expression
uzin2  |-  ( ( A  e.  ran  ZZ>=  /\  B  e.  ran  ZZ>= )  -> 
( A  i^i  B
)  e.  ran  ZZ>= )

Proof of Theorem uzin2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzf 9076 . . . 4  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5174 . . . 4  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
31, 2ax-mp 7 . . 3  |-  ZZ>=  Fn  ZZ
4 fvelrnb 5365 . . 3  |-  ( ZZ>=  Fn  ZZ  ->  ( A  e.  ran  ZZ>= 
<->  E. x  e.  ZZ  ( ZZ>= `  x )  =  A ) )
53, 4ax-mp 7 . 2  |-  ( A  e.  ran  ZZ>=  <->  E. x  e.  ZZ  ( ZZ>= `  x
)  =  A )
6 fvelrnb 5365 . . 3  |-  ( ZZ>=  Fn  ZZ  ->  ( B  e.  ran  ZZ>= 
<->  E. y  e.  ZZ  ( ZZ>= `  y )  =  B ) )
73, 6ax-mp 7 . 2  |-  ( B  e.  ran  ZZ>=  <->  E. y  e.  ZZ  ( ZZ>= `  y
)  =  B )
8 ineq1 3195 . . 3  |-  ( (
ZZ>= `  x )  =  A  ->  ( ( ZZ>=
`  x )  i^i  ( ZZ>= `  y )
)  =  ( A  i^i  ( ZZ>= `  y
) ) )
98eleq1d 2157 . 2  |-  ( (
ZZ>= `  x )  =  A  ->  ( (
( ZZ>= `  x )  i^i  ( ZZ>= `  y )
)  e.  ran  ZZ>=  <->  ( A  i^i  ( ZZ>= `  y )
)  e.  ran  ZZ>= ) )
10 ineq2 3196 . . 3  |-  ( (
ZZ>= `  y )  =  B  ->  ( A  i^i  ( ZZ>= `  y )
)  =  ( A  i^i  B ) )
1110eleq1d 2157 . 2  |-  ( (
ZZ>= `  y )  =  B  ->  ( ( A  i^i  ( ZZ>= `  y
) )  e.  ran  ZZ>=  <->  ( A  i^i  B )  e. 
ran  ZZ>= ) )
12 uzin 9105 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  x
)  i^i  ( ZZ>= `  y ) )  =  ( ZZ>= `  if (
x  <_  y , 
y ,  x ) ) )
13 simpr 109 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  y  e.  ZZ )
14 simpl 108 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  x  e.  ZZ )
15 zdcle 8877 . . . . 5  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  -> DECID  x  <_  y )
1613, 14, 15ifcldcd 3430 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  if ( x  <_ 
y ,  y ,  x )  e.  ZZ )
17 fnfvelrn 5445 . . . 4  |-  ( (
ZZ>=  Fn  ZZ  /\  if ( x  <_  y ,  y ,  x )  e.  ZZ )  -> 
( ZZ>= `  if (
x  <_  y , 
y ,  x ) )  e.  ran  ZZ>= )
183, 16, 17sylancr 406 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ZZ>= `  if (
x  <_  y , 
y ,  x ) )  e.  ran  ZZ>= )
1912, 18eqeltrd 2165 . 2  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  x
)  i^i  ( ZZ>= `  y ) )  e. 
ran  ZZ>= )
205, 7, 9, 11, 192gencl 2653 1  |-  ( ( A  e.  ran  ZZ>=  /\  B  e.  ran  ZZ>= )  -> 
( A  i^i  B
)  e.  ran  ZZ>= )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   E.wrex 2361    i^i cin 2999   ifcif 3397   ~Pcpw 3433   class class class wbr 3851   ran crn 4452    Fn wfn 5023   -->wf 5024   ` cfv 5028    <_ cle 7577   ZZcz 8804   ZZ>=cuz 9073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-inn 8477  df-n0 8728  df-z 8805  df-uz 9074
This theorem is referenced by:  rexanuz  10475
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