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Theorem fin23lem15 8501
Description: Lemma for fin23 8556. 
U is a monotone function. (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
fin23lem15  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( U `  A )  C_  ( U `  B )
)
Distinct variable groups:    t, i, u    A, i, u    U, i, u
Allowed substitution hints:    A( t)    B( u, t, i)    U( t)

Proof of Theorem fin23lem15
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5689 . . 3  |-  ( b  =  B  ->  ( U `  b )  =  ( U `  B ) )
21sseq1d 3381 . 2  |-  ( b  =  B  ->  (
( U `  b
)  C_  ( U `  B )  <->  ( U `  B )  C_  ( U `  B )
) )
3 fveq2 5689 . . 3  |-  ( b  =  a  ->  ( U `  b )  =  ( U `  a ) )
43sseq1d 3381 . 2  |-  ( b  =  a  ->  (
( U `  b
)  C_  ( U `  B )  <->  ( U `  a )  C_  ( U `  B )
) )
5 fveq2 5689 . . 3  |-  ( b  =  suc  a  -> 
( U `  b
)  =  ( U `
 suc  a )
)
65sseq1d 3381 . 2  |-  ( b  =  suc  a  -> 
( ( U `  b )  C_  ( U `  B )  <->  ( U `  suc  a
)  C_  ( U `  B ) ) )
7 fveq2 5689 . . 3  |-  ( b  =  A  ->  ( U `  b )  =  ( U `  A ) )
87sseq1d 3381 . 2  |-  ( b  =  A  ->  (
( U `  b
)  C_  ( U `  B )  <->  ( U `  A )  C_  ( U `  B )
) )
9 ssid 3373 . . 3  |-  ( U `
 B )  C_  ( U `  B )
109a1i 11 . 2  |-  ( B  e.  om  ->  ( U `  B )  C_  ( U `  B
) )
11 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 )
1211fin23lem13 8499 . . . 4  |-  ( a  e.  om  ->  ( U `  suc  a ) 
C_  ( U `  a ) )
1312ad2antrr 725 . . 3  |-  ( ( ( a  e.  om  /\  B  e.  om )  /\  B  C_  a )  ->  ( U `  suc  a )  C_  ( U `  a )
)
14 sstr2 3361 . . 3  |-  ( ( U `  suc  a
)  C_  ( U `  a )  ->  (
( U `  a
)  C_  ( U `  B )  ->  ( U `  suc  a ) 
C_  ( U `  B ) ) )
1513, 14syl 16 . 2  |-  ( ( ( a  e.  om  /\  B  e.  om )  /\  B  C_  a )  ->  ( ( U `
 a )  C_  ( U `  B )  ->  ( U `  suc  a )  C_  ( U `  B )
) )
162, 4, 6, 8, 10, 15findsg 6501 1  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( U `  A )  C_  ( U `  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2970    i^i cin 3325    C_ wss 3326   (/)c0 3635   ifcif 3789   U.cuni 4089   suc csuc 4719   ran crn 4839   ` cfv 5416    e. cmpt2 6091   omcom 6474  seq𝜔cseqom 6900
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 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-2nd 6576  df-recs 6830  df-rdg 6864  df-seqom 6901
This theorem is referenced by:  fin23lem16  8502
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