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Theorem fin23lem19 7962
Description: Lemma for fin23 8015. 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 7957 . . . 4  |-  ( A  e.  om  ->  ( U `  suc  A )  =  if ( ( ( t `  A
)  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) ) )
3 eqif 3598 . . . 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 188 . . 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 3361 . . . . 5  |-  ( ( U `  suc  A
)  i^i  ( t `  A ) )  =  ( ( t `  A )  i^i  ( U `  suc  A ) )
6 ineq2 3364 . . . . . . 7  |-  ( ( U `  suc  A
)  =  ( U `
 A )  -> 
( ( t `  A )  i^i  ( U `  suc  A ) )  =  ( ( t `  A )  i^i  ( U `  A ) ) )
76eqeq1d 2291 . . . . . 6  |-  ( ( U `  suc  A
)  =  ( U `
 A )  -> 
( ( ( t `
 A )  i^i  ( U `  suc  A ) )  =  (/)  <->  (
( t `  A
)  i^i  ( U `  A ) )  =  (/) ) )
87biimparc 473 . . . . 5  |-  ( ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/)  /\  ( U `  suc  A )  =  ( U `  A ) )  -> 
( ( t `  A )  i^i  ( U `  suc  A ) )  =  (/) )
95, 8syl5eq 2327 . . . 4  |-  ( ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/)  /\  ( U `  suc  A )  =  ( U `  A ) )  -> 
( ( U `  suc  A )  i^i  (
t `  A )
)  =  (/) )
10 inss1 3389 . . . . . 6  |-  ( ( t `  A )  i^i  ( U `  A ) )  C_  ( t `  A
)
11 sseq1 3199 . . . . . 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 224 . . . . 5  |-  ( ( U `  suc  A
)  =  ( ( t `  A )  i^i  ( U `  A ) )  -> 
( U `  suc  A )  C_  ( t `  A ) )
1312adantl 452 . . . 4  |-  ( ( -.  ( ( t `
 A )  i^i  ( U `  A
) )  =  (/)  /\  ( U `  suc  A )  =  ( ( t `  A )  i^i  ( U `  A ) ) )  ->  ( U `  suc  A )  C_  (
t `  A )
)
149, 13orim12i 502 . . 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 15 . 2  |-  ( A  e.  om  ->  (
( ( U `  suc  A )  i^i  (
t `  A )
)  =  (/)  \/  ( U `  suc  A ) 
C_  ( t `  A ) ) )
1615orcomd 377 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 357    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565   U.cuni 3827   suc csuc 4394   omcom 4656   ran crn 4690   ` cfv 5255    e. cmpt2 5860  seq𝜔cseqom 6459
This theorem is referenced by:  fin23lem20  7963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-recs 6388  df-rdg 6423  df-seqom 6460
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