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Theorem fin23lem20 8209
Description: Lemma for fin23 8261. 
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 6704 . . . 4  |-  U  Fn  om
3 peano2 4857 . . . 4  |-  ( A  e.  om  ->  suc  A  e.  om )
4 fnfvelrn 5859 . . . 4  |-  ( ( U  Fn  om  /\  suc  A  e.  om )  ->  ( U `  suc  A )  e.  ran  U
)
52, 3, 4sylancr 645 . . 3  |-  ( A  e.  om  ->  ( U `  suc  A )  e.  ran  U )
6 intss1 4057 . . 3  |-  ( ( U `  suc  A
)  e.  ran  U  ->  |^| ran  U  C_  ( U `  suc  A
) )
75, 6syl 16 . 2  |-  ( A  e.  om  ->  |^| ran  U 
C_  ( U `  suc  A ) )
81fin23lem19 8208 . 2  |-  ( A  e.  om  ->  (
( U `  suc  A )  C_  ( t `  A )  \/  (
( U `  suc  A )  i^i  ( t `
 A ) )  =  (/) ) )
9 sstr2 3347 . . 3  |-  ( |^| ran 
U  C_  ( U `  suc  A )  -> 
( ( U `  suc  A )  C_  (
t `  A )  ->  |^| ran  U  C_  ( t `  A
) ) )
10 ssdisj 3669 . . . 4  |-  ( (
|^| ran  U  C_  ( U `  suc  A )  /\  ( ( U `
 suc  A )  i^i  ( t `  A
) )  =  (/) )  ->  ( |^| ran  U  i^i  ( t `  A ) )  =  (/) )
1110ex 424 . . 3  |-  ( |^| ran 
U  C_  ( U `  suc  A )  -> 
( ( ( U `
 suc  A )  i^i  ( t `  A
) )  =  (/)  ->  ( |^| ran  U  i^i  ( t `  A
) )  =  (/) ) )
129, 11orim12d 812 . 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 58 1  |-  ( A  e.  om  ->  ( |^| ran  U  C_  (
t `  A )  \/  ( |^| ran  U  i^i  ( t `  A
) )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620   ifcif 3731   U.cuni 4007   |^|cint 4042   suc csuc 4575   omcom 4837   ran crn 4871    Fn wfn 5441   ` cfv 5446    e. cmpt2 6075  seq𝜔cseqom 6696
This theorem is referenced by:  fin23lem30  8214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-recs 6625  df-rdg 6660  df-seqom 6697
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