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Theorem fin23lem13 8767
Description: Lemma for fin23 8824. Each step of  U is a decrease. (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
fin23lem13  |-  ( A  e.  om  ->  ( U `  suc  A ) 
C_  ( U `  A ) )
Distinct variable groups:    t, i, u    A, i, u    U, i, u
Allowed substitution hints:    A( t)    U( t)

Proof of Theorem fin23lem13
StepHypRef Expression
1 fin23lem.a . . 3  |-  U  = seq𝜔 ( ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) ) ) ,  U. ran  t )
21fin23lem12 8766 . 2  |-  ( A  e.  om  ->  ( U `  suc  A )  =  if ( ( ( t `  A
)  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) ) )
3 sseq1 3455 . . 3  |-  ( ( U `  A )  =  if ( ( ( t `  A
)  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) )  ->  ( ( U `
 A )  C_  ( U `  A )  <-> 
if ( ( ( t `  A )  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) ) 
C_  ( U `  A ) ) )
4 sseq1 3455 . . 3  |-  ( ( ( t `  A
)  i^i  ( U `  A ) )  =  if ( ( ( t `  A )  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) )  ->  ( ( ( t `  A )  i^i  ( U `  A ) )  C_  ( U `  A )  <-> 
if ( ( ( t `  A )  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) ) 
C_  ( U `  A ) ) )
5 ssid 3453 . . 3  |-  ( U `
 A )  C_  ( U `  A )
6 inss2 3655 . . 3  |-  ( ( t `  A )  i^i  ( U `  A ) )  C_  ( U `  A )
73, 4, 5, 6keephyp 3947 . 2  |-  if ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/) ,  ( U `  A ) ,  ( ( t `
 A )  i^i  ( U `  A
) ) )  C_  ( U `  A )
82, 7syl6eqss 3484 1  |-  ( A  e.  om  ->  ( U `  suc  A ) 
C_  ( U `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1446    e. wcel 1889   _Vcvv 3047    i^i cin 3405    C_ wss 3406   (/)c0 3733   ifcif 3883   U.cuni 4201   ran crn 4838   suc csuc 5428   ` cfv 5585    |-> cmpt2 6297   omcom 6697  seq𝜔cseqom 7169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-seqom 7170
This theorem is referenced by:  fin23lem15  8769  fin23lem17  8773
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