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Theorem fin23lem19 8218
Description: Lemma for fin23 8271. 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 8213 . . . 4  |-  ( A  e.  om  ->  ( U `  suc  A )  =  if ( ( ( t `  A
)  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) ) )
3 eqif 3774 . . . 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 190 . . 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 3535 . . . . 5  |-  ( ( U `  suc  A
)  i^i  ( t `  A ) )  =  ( ( t `  A )  i^i  ( U `  suc  A ) )
6 ineq2 3538 . . . . . . 7  |-  ( ( U `  suc  A
)  =  ( U `
 A )  -> 
( ( t `  A )  i^i  ( U `  suc  A ) )  =  ( ( t `  A )  i^i  ( U `  A ) ) )
76eqeq1d 2446 . . . . . 6  |-  ( ( U `  suc  A
)  =  ( U `
 A )  -> 
( ( ( t `
 A )  i^i  ( U `  suc  A ) )  =  (/)  <->  (
( t `  A
)  i^i  ( U `  A ) )  =  (/) ) )
87biimparc 475 . . . . 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 3563 . . . . . 6  |-  ( ( t `  A )  i^i  ( U `  A ) )  C_  ( t `  A
)
11 sseq1 3371 . . . . . 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 226 . . . . 5  |-  ( ( U `  suc  A
)  =  ( ( t `  A )  i^i  ( U `  A ) )  -> 
( U `  suc  A )  C_  ( t `  A ) )
1312adantl 454 . . . 4  |-  ( ( -.  ( ( t `
 A )  i^i  ( U `  A
) )  =  (/)  /\  ( U `  suc  A )  =  ( ( t `  A )  i^i  ( U `  A ) ) )  ->  ( U `  suc  A )  C_  (
t `  A )
)
149, 13orim12i 504 . . 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 16 . 2  |-  ( A  e.  om  ->  (
( ( U `  suc  A )  i^i  (
t `  A )
)  =  (/)  \/  ( U `  suc  A ) 
C_  ( t `  A ) ) )
1615orcomd 379 1  |-  ( A  e.  om  ->  (
( U `  suc  A )  C_  ( t `  A )  \/  (
( U `  suc  A )  i^i  ( t `
 A ) )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   ifcif 3741   U.cuni 4017   suc csuc 4585   omcom 4847   ran crn 4881   ` cfv 5456    e. cmpt2 6085  seq𝜔cseqom 6706
This theorem is referenced by:  fin23lem20  8219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-2nd 6352  df-recs 6635  df-rdg 6670  df-seqom 6707
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