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Theorem fin23lem20 8798
Description: Lemma for fin23 8850. 
X is either contained in or disjoint from all input sets. (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
fin23lem20  |-  ( A  e.  om  ->  ( |^| ran  U  C_  (
t `  A )  \/  ( |^| ran  U  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 fin23lem20
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 )
21fnseqom 7203 . . . 4  |-  U  Fn  om
3 peano2 6745 . . . 4  |-  ( A  e.  om  ->  suc  A  e.  om )
4 fnfvelrn 6047 . . . 4  |-  ( ( U  Fn  om  /\  suc  A  e.  om )  ->  ( U `  suc  A )  e.  ran  U
)
52, 3, 4sylancr 674 . . 3  |-  ( A  e.  om  ->  ( U `  suc  A )  e.  ran  U )
6 intss1 4263 . . 3  |-  ( ( U `  suc  A
)  e.  ran  U  ->  |^| ran  U  C_  ( U `  suc  A
) )
75, 6syl 17 . 2  |-  ( A  e.  om  ->  |^| ran  U 
C_  ( U `  suc  A ) )
81fin23lem19 8797 . 2  |-  ( A  e.  om  ->  (
( U `  suc  A )  C_  ( t `  A )  \/  (
( U `  suc  A )  i^i  ( t `
 A ) )  =  (/) ) )
9 sstr2 3451 . . 3  |-  ( |^| ran 
U  C_  ( U `  suc  A )  -> 
( ( U `  suc  A )  C_  (
t `  A )  ->  |^| ran  U  C_  ( t `  A
) ) )
10 ssdisj 3826 . . . 4  |-  ( (
|^| ran  U  C_  ( U `  suc  A )  /\  ( ( U `
 suc  A )  i^i  ( t `  A
) )  =  (/) )  ->  ( |^| ran  U  i^i  ( t `  A ) )  =  (/) )
1110ex 440 . . 3  |-  ( |^| ran 
U  C_  ( U `  suc  A )  -> 
( ( ( U `
 suc  A )  i^i  ( t `  A
) )  =  (/)  ->  ( |^| ran  U  i^i  ( t `  A
) )  =  (/) ) )
129, 11orim12d 854 . 2  |-  ( |^| ran 
U  C_  ( U `  suc  A )  -> 
( ( ( U `
 suc  A )  C_  ( t `  A
)  \/  ( ( U `  suc  A
)  i^i  ( t `  A ) )  =  (/) )  ->  ( |^| ran 
U  C_  ( t `  A )  \/  ( |^| ran  U  i^i  (
t `  A )
)  =  (/) ) ) )
137, 8, 12sylc 62 1  |-  ( A  e.  om  ->  ( |^| ran  U  C_  (
t `  A )  \/  ( |^| ran  U  i^i  ( t `  A
) )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 374    = wceq 1455    e. wcel 1898   _Vcvv 3057    i^i cin 3415    C_ wss 3416   (/)c0 3743   ifcif 3893   U.cuni 4212   |^|cint 4248   ran crn 4857   suc csuc 5448    Fn wfn 5600   ` cfv 5605    |-> cmpt2 6322   omcom 6724  seq𝜔cseqom 7195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-seqom 7196
This theorem is referenced by:  fin23lem30  8803
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