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Theorem uzin2 12153
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 10496 . . . 4  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5594 . . . 4  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
31, 2ax-mp 5 . . 3  |-  ZZ>=  Fn  ZZ
4 fvelrnb 5777 . . 3  |-  ( ZZ>=  Fn  ZZ  ->  ( A  e.  ran  ZZ>= 
<->  E. x  e.  ZZ  ( ZZ>= `  x )  =  A ) )
53, 4ax-mp 5 . 2  |-  ( A  e.  ran  ZZ>=  <->  E. x  e.  ZZ  ( ZZ>= `  x
)  =  A )
6 fvelrnb 5777 . . 3  |-  ( ZZ>=  Fn  ZZ  ->  ( B  e.  ran  ZZ>= 
<->  E. y  e.  ZZ  ( ZZ>= `  y )  =  B ) )
73, 6ax-mp 5 . 2  |-  ( B  e.  ran  ZZ>=  <->  E. y  e.  ZZ  ( ZZ>= `  y
)  =  B )
8 ineq1 3537 . . 3  |-  ( (
ZZ>= `  x )  =  A  ->  ( ( ZZ>=
`  x )  i^i  ( ZZ>= `  y )
)  =  ( A  i^i  ( ZZ>= `  y
) ) )
98eleq1d 2504 . 2  |-  ( (
ZZ>= `  x )  =  A  ->  ( (
( ZZ>= `  x )  i^i  ( ZZ>= `  y )
)  e.  ran  ZZ>=  <->  ( A  i^i  ( ZZ>= `  y )
)  e.  ran  ZZ>= ) )
10 ineq2 3538 . . 3  |-  ( (
ZZ>= `  y )  =  B  ->  ( A  i^i  ( ZZ>= `  y )
)  =  ( A  i^i  B ) )
1110eleq1d 2504 . 2  |-  ( (
ZZ>= `  y )  =  B  ->  ( ( A  i^i  ( ZZ>= `  y
) )  e.  ran  ZZ>=  <->  ( A  i^i  B )  e. 
ran  ZZ>= ) )
12 uzin 10523 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  x
)  i^i  ( ZZ>= `  y ) )  =  ( ZZ>= `  if (
x  <_  y , 
y ,  x ) ) )
13 ifcl 3777 . . . . 5  |-  ( ( y  e.  ZZ  /\  x  e.  ZZ )  ->  if ( x  <_ 
y ,  y ,  x )  e.  ZZ )
1413ancoms 441 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  if ( x  <_ 
y ,  y ,  x )  e.  ZZ )
15 fnfvelrn 5870 . . . 4  |-  ( (
ZZ>=  Fn  ZZ  /\  if ( x  <_  y ,  y ,  x )  e.  ZZ )  -> 
( ZZ>= `  if (
x  <_  y , 
y ,  x ) )  e.  ran  ZZ>= )
163, 14, 15sylancr 646 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ZZ>= `  if (
x  <_  y , 
y ,  x ) )  e.  ran  ZZ>= )
1712, 16eqeltrd 2512 . 2  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  x
)  i^i  ( ZZ>= `  y ) )  e. 
ran  ZZ>= )
185, 7, 9, 11, 172gencl 2987 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    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    i^i cin 3321   ifcif 3741   ~Pcpw 3801   class class class wbr 4215   ran crn 4882    Fn wfn 5452   -->wf 5453   ` cfv 5457    <_ cle 9126   ZZcz 10287   ZZ>=cuz 10493
This theorem is referenced by:  rexanuz  12154  zfbas  17933  heibor1lem  26532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-pre-lttri 9069  ax-pre-lttrn 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-neg 9299  df-z 10288  df-uz 10494
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