MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem20 Unicode version

Theorem fin23lem20 8150
Description: Lemma for fin23 8202. 
X is either contained in or disjoint from all input sets. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Hypothesis
Ref Expression
fin23lem.a  |-  U  = seq𝜔 ( ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) ) ) ,  U. ran  t )
Assertion
Ref Expression
fin23lem20  |-  ( A  e.  om  ->  ( |^| ran  U  C_  (
t `  A )  \/  ( |^| ran  U  i^i  ( t `  A
) )  =  (/) ) )
Distinct variable groups:    t, i, u    A, i, u    U, i, u
Allowed substitution hints:    A( t)    U( t)

Proof of Theorem fin23lem20
StepHypRef Expression
1 fin23lem.a . . . . 5  |-  U  = seq𝜔 ( ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) ) ) ,  U. ran  t )
21fnseqom 6648 . . . 4  |-  U  Fn  om
3 peano2 4805 . . . 4  |-  ( A  e.  om  ->  suc  A  e.  om )
4 fnfvelrn 5806 . . . 4  |-  ( ( U  Fn  om  /\  suc  A  e.  om )  ->  ( U `  suc  A )  e.  ran  U
)
52, 3, 4sylancr 645 . . 3  |-  ( A  e.  om  ->  ( U `  suc  A )  e.  ran  U )
6 intss1 4007 . . 3  |-  ( ( U `  suc  A
)  e.  ran  U  ->  |^| ran  U  C_  ( U `  suc  A
) )
75, 6syl 16 . 2  |-  ( A  e.  om  ->  |^| ran  U 
C_  ( U `  suc  A ) )
81fin23lem19 8149 . 2  |-  ( A  e.  om  ->  (
( U `  suc  A )  C_  ( t `  A )  \/  (
( U `  suc  A )  i^i  ( t `
 A ) )  =  (/) ) )
9 sstr2 3298 . . 3  |-  ( |^| ran 
U  C_  ( U `  suc  A )  -> 
( ( U `  suc  A )  C_  (
t `  A )  ->  |^| ran  U  C_  ( t `  A
) ) )
10 ssdisj 3620 . . . 4  |-  ( (
|^| ran  U  C_  ( U `  suc  A )  /\  ( ( U `
 suc  A )  i^i  ( t `  A
) )  =  (/) )  ->  ( |^| ran  U  i^i  ( t `  A ) )  =  (/) )
1110ex 424 . . 3  |-  ( |^| ran 
U  C_  ( U `  suc  A )  -> 
( ( ( U `
 suc  A )  i^i  ( t `  A
) )  =  (/)  ->  ( |^| ran  U  i^i  ( t `  A
) )  =  (/) ) )
129, 11orim12d 812 . 2  |-  ( |^| ran 
U  C_  ( U `  suc  A )  -> 
( ( ( U `
 suc  A )  C_  ( t `  A
)  \/  ( ( U `  suc  A
)  i^i  ( t `  A ) )  =  (/) )  ->  ( |^| ran 
U  C_  ( t `  A )  \/  ( |^| ran  U  i^i  (
t `  A )
)  =  (/) ) ) )
137, 8, 12sylc 58 1  |-  ( A  e.  om  ->  ( |^| ran  U  C_  (
t `  A )  \/  ( |^| ran  U  i^i  ( t `  A
) )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    = wceq 1649    e. wcel 1717   _Vcvv 2899    i^i cin 3262    C_ wss 3263   (/)c0 3571   ifcif 3682   U.cuni 3957   |^|cint 3992   suc csuc 4524   omcom 4785   ran crn 4819    Fn wfn 5389   ` cfv 5394    e. cmpt2 6022  seq𝜔cseqom 6640
This theorem is referenced by:  fin23lem30  8155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-recs 6569  df-rdg 6604  df-seqom 6641
  Copyright terms: Public domain W3C validator