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Theorem fin23lem15 8214
Description: Lemma for fin23 8269. 
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 5728 . . 3  |-  ( b  =  B  ->  ( U `  b )  =  ( U `  B ) )
21sseq1d 3375 . 2  |-  ( b  =  B  ->  (
( U `  b
)  C_  ( U `  B )  <->  ( U `  B )  C_  ( U `  B )
) )
3 fveq2 5728 . . 3  |-  ( b  =  a  ->  ( U `  b )  =  ( U `  a ) )
43sseq1d 3375 . 2  |-  ( b  =  a  ->  (
( U `  b
)  C_  ( U `  B )  <->  ( U `  a )  C_  ( U `  B )
) )
5 fveq2 5728 . . 3  |-  ( b  =  suc  a  -> 
( U `  b
)  =  ( U `
 suc  a )
)
65sseq1d 3375 . 2  |-  ( b  =  suc  a  -> 
( ( U `  b )  C_  ( U `  B )  <->  ( U `  suc  a
)  C_  ( U `  B ) ) )
7 fveq2 5728 . . 3  |-  ( b  =  A  ->  ( U `  b )  =  ( U `  A ) )
87sseq1d 3375 . 2  |-  ( b  =  A  ->  (
( U `  b
)  C_  ( U `  B )  <->  ( U `  A )  C_  ( U `  B )
) )
9 ssid 3367 . . 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 8212 . . . 4  |-  ( a  e.  om  ->  ( U `  suc  a ) 
C_  ( U `  a ) )
1312ad2antrr 707 . . 3  |-  ( ( ( a  e.  om  /\  B  e.  om )  /\  B  C_  a )  ->  ( U `  suc  a )  C_  ( U `  a )
)
14 sstr2 3355 . . 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 4872 1  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( U `  A )  C_  ( U `  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319    C_ wss 3320   (/)c0 3628   ifcif 3739   U.cuni 4015   suc csuc 4583   omcom 4845   ran crn 4879   ` cfv 5454    e. cmpt2 6083  seq𝜔cseqom 6704
This theorem is referenced by:  fin23lem16  8215
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-recs 6633  df-rdg 6668  df-seqom 6705
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