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Theorem uzin2 13126
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 11074 . . . 4  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5722 . . . 4  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
31, 2ax-mp 5 . . 3  |-  ZZ>=  Fn  ZZ
4 fvelrnb 5906 . . 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 5906 . . 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 3686 . . 3  |-  ( (
ZZ>= `  x )  =  A  ->  ( ( ZZ>=
`  x )  i^i  ( ZZ>= `  y )
)  =  ( A  i^i  ( ZZ>= `  y
) ) )
98eleq1d 2529 . 2  |-  ( (
ZZ>= `  x )  =  A  ->  ( (
( ZZ>= `  x )  i^i  ( ZZ>= `  y )
)  e.  ran  ZZ>=  <->  ( A  i^i  ( ZZ>= `  y )
)  e.  ran  ZZ>= ) )
10 ineq2 3687 . . 3  |-  ( (
ZZ>= `  y )  =  B  ->  ( A  i^i  ( ZZ>= `  y )
)  =  ( A  i^i  B ) )
1110eleq1d 2529 . 2  |-  ( (
ZZ>= `  y )  =  B  ->  ( ( A  i^i  ( ZZ>= `  y
) )  e.  ran  ZZ>=  <->  ( A  i^i  B )  e. 
ran  ZZ>= ) )
12 uzin 11103 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  x
)  i^i  ( ZZ>= `  y ) )  =  ( ZZ>= `  if (
x  <_  y , 
y ,  x ) ) )
13 ifcl 3974 . . . . 5  |-  ( ( y  e.  ZZ  /\  x  e.  ZZ )  ->  if ( x  <_ 
y ,  y ,  x )  e.  ZZ )
1413ancoms 453 . . . 4  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  if ( x  <_ 
y ,  y ,  x )  e.  ZZ )
15 fnfvelrn 6009 . . . 4  |-  ( (
ZZ>=  Fn  ZZ  /\  if ( x  <_  y ,  y ,  x )  e.  ZZ )  -> 
( ZZ>= `  if (
x  <_  y , 
y ,  x ) )  e.  ran  ZZ>= )
163, 14, 15sylancr 663 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ZZ>= `  if (
x  <_  y , 
y ,  x ) )  e.  ran  ZZ>= )
1712, 16eqeltrd 2548 . 2  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  x
)  i^i  ( ZZ>= `  y ) )  e. 
ran  ZZ>= )
185, 7, 9, 11, 172gencl 3137 1  |-  ( ( A  e.  ran  ZZ>=  /\  B  e.  ran  ZZ>= )  -> 
( A  i^i  B
)  e.  ran  ZZ>= )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   E.wrex 2808    i^i cin 3468   ifcif 3932   ~Pcpw 4003   class class class wbr 4440   ran crn 4993    Fn wfn 5574   -->wf 5575   ` cfv 5579    <_ cle 9618   ZZcz 10853   ZZ>=cuz 11071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-neg 9797  df-z 10854  df-uz 11072
This theorem is referenced by:  rexanuz  13127  zfbas  20125  heibor1lem  29759
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