MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem19 Structured version   Unicode version

Theorem fin23lem19 8764
Description: Lemma for fin23 8817. The first set in  U to see an input set is either contained in it or disjoint from it. (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
fin23lem19  |-  ( A  e.  om  ->  (
( U `  suc  A )  C_  ( t `  A )  \/  (
( U `  suc  A )  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 fin23lem19
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 )
21fin23lem12 8759 . . . 4  |-  ( A  e.  om  ->  ( U `  suc  A )  =  if ( ( ( t `  A
)  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) ) )
3 eqif 3953 . . . 4  |-  ( ( U `  suc  A
)  =  if ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/) ,  ( U `  A ) ,  ( ( t `
 A )  i^i  ( U `  A
) ) )  <->  ( (
( ( t `  A )  i^i  ( U `  A )
)  =  (/)  /\  ( U `  suc  A )  =  ( U `  A ) )  \/  ( -.  ( ( t `  A )  i^i  ( U `  A ) )  =  (/)  /\  ( U `  suc  A )  =  ( ( t `  A
)  i^i  ( U `  A ) ) ) ) )
42, 3sylib 199 . . 3  |-  ( A  e.  om  ->  (
( ( ( t `
 A )  i^i  ( U `  A
) )  =  (/)  /\  ( U `  suc  A )  =  ( U `
 A ) )  \/  ( -.  (
( t `  A
)  i^i  ( U `  A ) )  =  (/)  /\  ( U `  suc  A )  =  ( ( t `  A
)  i^i  ( U `  A ) ) ) ) )
5 incom 3661 . . . . 5  |-  ( ( U `  suc  A
)  i^i  ( t `  A ) )  =  ( ( t `  A )  i^i  ( U `  suc  A ) )
6 ineq2 3664 . . . . . . 7  |-  ( ( U `  suc  A
)  =  ( U `
 A )  -> 
( ( t `  A )  i^i  ( U `  suc  A ) )  =  ( ( t `  A )  i^i  ( U `  A ) ) )
76eqeq1d 2431 . . . . . 6  |-  ( ( U `  suc  A
)  =  ( U `
 A )  -> 
( ( ( t `
 A )  i^i  ( U `  suc  A ) )  =  (/)  <->  (
( t `  A
)  i^i  ( U `  A ) )  =  (/) ) )
87biimparc 489 . . . . 5  |-  ( ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/)  /\  ( U `  suc  A )  =  ( U `  A ) )  -> 
( ( t `  A )  i^i  ( U `  suc  A ) )  =  (/) )
95, 8syl5eq 2482 . . . 4  |-  ( ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/)  /\  ( U `  suc  A )  =  ( U `  A ) )  -> 
( ( U `  suc  A )  i^i  (
t `  A )
)  =  (/) )
10 inss1 3688 . . . . . 6  |-  ( ( t `  A )  i^i  ( U `  A ) )  C_  ( t `  A
)
11 sseq1 3491 . . . . . 6  |-  ( ( U `  suc  A
)  =  ( ( t `  A )  i^i  ( U `  A ) )  -> 
( ( U `  suc  A )  C_  (
t `  A )  <->  ( ( t `  A
)  i^i  ( U `  A ) )  C_  ( t `  A
) ) )
1210, 11mpbiri 236 . . . . 5  |-  ( ( U `  suc  A
)  =  ( ( t `  A )  i^i  ( U `  A ) )  -> 
( U `  suc  A )  C_  ( t `  A ) )
1312adantl 467 . . . 4  |-  ( ( -.  ( ( t `
 A )  i^i  ( U `  A
) )  =  (/)  /\  ( U `  suc  A )  =  ( ( t `  A )  i^i  ( U `  A ) ) )  ->  ( U `  suc  A )  C_  (
t `  A )
)
149, 13orim12i 518 . . 3  |-  ( ( ( ( ( t `
 A )  i^i  ( U `  A
) )  =  (/)  /\  ( U `  suc  A )  =  ( U `
 A ) )  \/  ( -.  (
( t `  A
)  i^i  ( U `  A ) )  =  (/)  /\  ( U `  suc  A )  =  ( ( t `  A
)  i^i  ( U `  A ) ) ) )  ->  ( (
( U `  suc  A )  i^i  ( t `
 A ) )  =  (/)  \/  ( U `  suc  A ) 
C_  ( t `  A ) ) )
154, 14syl 17 . 2  |-  ( A  e.  om  ->  (
( ( U `  suc  A )  i^i  (
t `  A )
)  =  (/)  \/  ( U `  suc  A ) 
C_  ( t `  A ) ) )
1615orcomd 389 1  |-  ( A  e.  om  ->  (
( U `  suc  A )  C_  ( t `  A )  \/  (
( U `  suc  A )  i^i  ( t `
 A ) )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    i^i cin 3441    C_ wss 3442   (/)c0 3767   ifcif 3915   U.cuni 4222   ran crn 4855   suc csuc 5444   ` cfv 5601    |-> cmpt2 6307   omcom 6706  seq𝜔cseqom 7172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-seqom 7173
This theorem is referenced by:  fin23lem20  8765
  Copyright terms: Public domain W3C validator