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Theorem elfz2 11081
Description: Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show  M  e.  ZZ and  N  e.  ZZ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
elfz2  |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) ) )

Proof of Theorem elfz2
StepHypRef Expression
1 anass 632 . 2  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) )
2 df-3an 939 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ ) )
32anbi1i 678 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) )  <->  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N
) ) )
4 elfz1 11079 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <-> 
( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N ) ) )
5 3anass 941 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N )  <->  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) )
6 ibar 492 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
75, 6syl5bb 250 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N
)  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
84, 7bitrd 246 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <-> 
( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
9 fzf 11078 . . . . . . 7  |-  ... :
( ZZ  X.  ZZ )
--> ~P ZZ
109fdmi 5625 . . . . . 6  |-  dom  ...  =  ( ZZ  X.  ZZ )
1110ndmov 6260 . . . . 5  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N )  =  (/) )
1211eleq2d 2509 . . . 4  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N
)  <->  K  e.  (/) ) )
13 noel 3617 . . . . . 6  |-  -.  K  e.  (/)
1413pm2.21i 126 . . . . 5  |-  ( K  e.  (/)  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
15 simpl 445 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
1614, 15pm5.21ni 343 . . . 4  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  (/) 
<->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
1712, 16bitrd 246 . . 3  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N
)  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
188, 17pm2.61i 159 . 2  |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) )
191, 3, 183bitr4ri 271 1  |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1727   (/)c0 3613   ~Pcpw 3823   class class class wbr 4237    X. cxp 4905  (class class class)co 6110    <_ cle 9152   ZZcz 10313   ...cfz 11074
This theorem is referenced by:  elfz4  11083  elfzuzb  11084  fzind2  11229  fzp1nel  25241  fprodntriv  25299  fprodeq0  25330  preduz  25506  fmul01lt1lem1  27728  fmul01lt1lem2  27729  stoweidlem3  27766  stoweidlem34  27797  stoweidlem51  27814  elfzmlbm  28153  elfzmlbp  28154  elfzelfzelfz  28156  elfz0fzfz0  28161  fzmmmeqm  28162  fz0fzelfz0  28165  fz0fzdiffz0  28166  swrdswrdlem  28256  swrdswrd  28257  swrdccatin12lem3a  28266  swrdccatin12lem3b  28267  swrdccatin2  28268  swrdccatin12lem3  28270  swrdccatin12  28272  swrdccat3  28273  2cshw1lem1  28306  2cshw2lem1  28310  2cshw2lem2  28311  cshweqdif2  28328  cshweqdif2s  28329
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 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-neg 9325  df-z 10314  df-fz 11075
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