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Theorem fin23lem19 8733
Description: Lemma for fin23 8786. 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 8728 . . . 4  |-  ( A  e.  om  ->  ( U `  suc  A )  =  if ( ( ( t `  A
)  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) ) )
3 eqif 3982 . . . 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 3687 . . . . 5  |-  ( ( U `  suc  A
)  i^i  ( t `  A ) )  =  ( ( t `  A )  i^i  ( U `  suc  A ) )
6 ineq2 3690 . . . . . . 7  |-  ( ( U `  suc  A
)  =  ( U `
 A )  -> 
( ( t `  A )  i^i  ( U `  suc  A ) )  =  ( ( t `  A )  i^i  ( U `  A ) ) )
76eqeq1d 2459 . . . . . 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 2510 . . . 4  |-  ( ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/)  /\  ( U `  suc  A )  =  ( U `  A ) )  -> 
( ( U `  suc  A )  i^i  (
t `  A )
)  =  (/) )
10 inss1 3714 . . . . . 6  |-  ( ( t `  A )  i^i  ( U `  A ) )  C_  ( t `  A
)
11 sseq1 3520 . . . . . 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 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   ifcif 3944   U.cuni 4251   suc csuc 4889   ran crn 5009   ` cfv 5594    |-> cmpt2 6298   omcom 6699  seq𝜔cseqom 7130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-seqom 7131
This theorem is referenced by:  fin23lem20  8734
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