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Theorem fin23lem12 8702
Description: The beginning of the proof that every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets).

This first section is dedicated to the construction of  U and its intersection. First, the value of  U at a successor. (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
fin23lem12  |-  ( A  e.  om  ->  ( U `  suc  A )  =  if ( ( ( t `  A
)  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) ) )
Distinct variable groups:    t, i, u    A, i, u    U, i, u
Allowed substitution hints:    A( t)    U( t)

Proof of Theorem fin23lem12
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 )
21seqomsuc 7114 . 2  |-  ( A  e.  om  ->  ( U `  suc  A )  =  ( A ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `  i )  i^i  u
)  =  (/) ,  u ,  ( ( t `
 i )  i^i  u ) ) ) ( U `  A
) ) )
3 fvex 5869 . . 3  |-  ( U `
 A )  e. 
_V
4 fveq2 5859 . . . . . . 7  |-  ( i  =  A  ->  (
t `  i )  =  ( t `  A ) )
54ineq1d 3694 . . . . . 6  |-  ( i  =  A  ->  (
( t `  i
)  i^i  u )  =  ( ( t `
 A )  i^i  u ) )
65eqeq1d 2464 . . . . 5  |-  ( i  =  A  ->  (
( ( t `  i )  i^i  u
)  =  (/)  <->  ( (
t `  A )  i^i  u )  =  (/) ) )
76, 5ifbieq2d 3959 . . . 4  |-  ( i  =  A  ->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) )  =  if ( ( ( t `  A
)  i^i  u )  =  (/) ,  u ,  ( ( t `  A )  i^i  u
) ) )
8 ineq2 3689 . . . . . 6  |-  ( u  =  ( U `  A )  ->  (
( t `  A
)  i^i  u )  =  ( ( t `
 A )  i^i  ( U `  A
) ) )
98eqeq1d 2464 . . . . 5  |-  ( u  =  ( U `  A )  ->  (
( ( t `  A )  i^i  u
)  =  (/)  <->  ( (
t `  A )  i^i  ( U `  A
) )  =  (/) ) )
10 id 22 . . . . 5  |-  ( u  =  ( U `  A )  ->  u  =  ( U `  A ) )
119, 10, 8ifbieq12d 3961 . . . 4  |-  ( u  =  ( U `  A )  ->  if ( ( ( t `
 A )  i^i  u )  =  (/) ,  u ,  ( ( t `  A )  i^i  u ) )  =  if ( ( ( t `  A
)  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) ) )
12 eqid 2462 . . . 4  |-  ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `  i
)  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u
) ) )  =  ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) ) )
133inex2 4584 . . . . 5  |-  ( ( t `  A )  i^i  ( U `  A ) )  e. 
_V
143, 13ifex 4003 . . . 4  |-  if ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/) ,  ( U `  A ) ,  ( ( t `
 A )  i^i  ( U `  A
) ) )  e. 
_V
157, 11, 12, 14ovmpt2 6415 . . 3  |-  ( ( A  e.  om  /\  ( U `  A )  e.  _V )  -> 
( A ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `  i
)  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u
) ) ) ( U `  A ) )  =  if ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/) ,  ( U `  A ) ,  ( ( t `
 A )  i^i  ( U `  A
) ) ) )
163, 15mpan2 671 . 2  |-  ( A  e.  om  ->  ( A ( i  e. 
om ,  u  e. 
_V  |->  if ( ( ( t `  i
)  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u
) ) ) ( U `  A ) )  =  if ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/) ,  ( U `  A ) ,  ( ( t `
 A )  i^i  ( U `  A
) ) ) )
172, 16eqtrd 2503 1  |-  ( A  e.  om  ->  ( U `  suc  A )  =  if ( ( ( t `  A
)  i^i  ( U `  A ) )  =  (/) ,  ( U `  A ) ,  ( ( t `  A
)  i^i  ( U `  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3108    i^i cin 3470   (/)c0 3780   ifcif 3934   U.cuni 4240   suc csuc 4875   ran crn 4995   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   omcom 6673  seq𝜔cseqom 7104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-2nd 6777  df-recs 7034  df-rdg 7068  df-seqom 7105
This theorem is referenced by:  fin23lem13  8703  fin23lem14  8704  fin23lem19  8707
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