MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uzin2 Structured version   Unicode version

Theorem uzin2 13156
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 11093 . . . 4  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5721 . . . 4  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
31, 2ax-mp 5 . . 3  |-  ZZ>=  Fn  ZZ
4 fvelrnb 5905 . . 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 5905 . . 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 3678 . . 3  |-  ( (
ZZ>= `  x )  =  A  ->  ( ( ZZ>=
`  x )  i^i  ( ZZ>= `  y )
)  =  ( A  i^i  ( ZZ>= `  y
) ) )
98eleq1d 2512 . 2  |-  ( (
ZZ>= `  x )  =  A  ->  ( (
( ZZ>= `  x )  i^i  ( ZZ>= `  y )
)  e.  ran  ZZ>=  <->  ( A  i^i  ( ZZ>= `  y )
)  e.  ran  ZZ>= ) )
10 ineq2 3679 . . 3  |-  ( (
ZZ>= `  y )  =  B  ->  ( A  i^i  ( ZZ>= `  y )
)  =  ( A  i^i  B ) )
1110eleq1d 2512 . 2  |-  ( (
ZZ>= `  y )  =  B  ->  ( ( A  i^i  ( ZZ>= `  y
) )  e.  ran  ZZ>=  <->  ( A  i^i  B )  e. 
ran  ZZ>= ) )
12 uzin 11122 . . 3  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  x
)  i^i  ( ZZ>= `  y ) )  =  ( ZZ>= `  if (
x  <_  y , 
y ,  x ) ) )
13 ifcl 3968 . . . . 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 6013 . . . 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 2531 . 2  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( ZZ>= `  x
)  i^i  ( ZZ>= `  y ) )  e. 
ran  ZZ>= )
185, 7, 9, 11, 172gencl 3126 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 1383    e. wcel 1804   E.wrex 2794    i^i cin 3460   ifcif 3926   ~Pcpw 3997   class class class wbr 4437   ran crn 4990    Fn wfn 5573   -->wf 5574   ` cfv 5578    <_ cle 9632   ZZcz 10870   ZZ>=cuz 11090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-pre-lttri 9569  ax-pre-lttrn 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-neg 9813  df-z 10871  df-uz 11091
This theorem is referenced by:  rexanuz  13157  zfbas  20270  heibor1lem  30280
  Copyright terms: Public domain W3C validator