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Theorem fin23lem19 8526
Description: Lemma for fin23 8579. 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 8521 . . . 4  |-  ( A  e.  om  ->  ( U `  suc  A )  =  if ( ( ( t `  A
)  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) ) )
3 eqif 3848 . . . 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 196 . . 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 3564 . . . . 5  |-  ( ( U `  suc  A
)  i^i  ( t `  A ) )  =  ( ( t `  A )  i^i  ( U `  suc  A ) )
6 ineq2 3567 . . . . . . 7  |-  ( ( U `  suc  A
)  =  ( U `
 A )  -> 
( ( t `  A )  i^i  ( U `  suc  A ) )  =  ( ( t `  A )  i^i  ( U `  A ) ) )
76eqeq1d 2451 . . . . . 6  |-  ( ( U `  suc  A
)  =  ( U `
 A )  -> 
( ( ( t `
 A )  i^i  ( U `  suc  A ) )  =  (/)  <->  (
( t `  A
)  i^i  ( U `  A ) )  =  (/) ) )
87biimparc 487 . . . . 5  |-  ( ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/)  /\  ( U `  suc  A )  =  ( U `  A ) )  -> 
( ( t `  A )  i^i  ( U `  suc  A ) )  =  (/) )
95, 8syl5eq 2487 . . . 4  |-  ( ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/)  /\  ( U `  suc  A )  =  ( U `  A ) )  -> 
( ( U `  suc  A )  i^i  (
t `  A )
)  =  (/) )
10 inss1 3591 . . . . . 6  |-  ( ( t `  A )  i^i  ( U `  A ) )  C_  ( t `  A
)
11 sseq1 3398 . . . . . 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 233 . . . . 5  |-  ( ( U `  suc  A
)  =  ( ( t `  A )  i^i  ( U `  A ) )  -> 
( U `  suc  A )  C_  ( t `  A ) )
1312adantl 466 . . . 4  |-  ( ( -.  ( ( t `
 A )  i^i  ( U `  A
) )  =  (/)  /\  ( U `  suc  A )  =  ( ( t `  A )  i^i  ( U `  A ) ) )  ->  ( U `  suc  A )  C_  (
t `  A )
)
149, 13orim12i 516 . . 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 388 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 368    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2993    i^i cin 3348    C_ wss 3349   (/)c0 3658   ifcif 3812   U.cuni 4112   suc csuc 4742   ran crn 4862   ` cfv 5439    e. cmpt2 6114   omcom 6497  seq𝜔cseqom 6923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-2nd 6599  df-recs 6853  df-rdg 6887  df-seqom 6924
This theorem is referenced by:  fin23lem20  8527
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