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Theorem fin23lem12 8242
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 6743 . 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 5771 . . 3  |-  ( U `
 A )  e. 
_V
4 fveq2 5757 . . . . . . 7  |-  ( i  =  A  ->  (
t `  i )  =  ( t `  A ) )
54ineq1d 3527 . . . . . 6  |-  ( i  =  A  ->  (
( t `  i
)  i^i  u )  =  ( ( t `
 A )  i^i  u ) )
65eqeq1d 2450 . . . . 5  |-  ( i  =  A  ->  (
( ( t `  i )  i^i  u
)  =  (/)  <->  ( (
t `  A )  i^i  u )  =  (/) ) )
76, 5ifbieq2d 3783 . . . 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 3522 . . . . . 6  |-  ( u  =  ( U `  A )  ->  (
( t `  A
)  i^i  u )  =  ( ( t `
 A )  i^i  ( U `  A
) ) )
98eqeq1d 2450 . . . . 5  |-  ( u  =  ( U `  A )  ->  (
( ( t `  A )  i^i  u
)  =  (/)  <->  ( (
t `  A )  i^i  ( U `  A
) )  =  (/) ) )
10 id 21 . . . . 5  |-  ( u  =  ( U `  A )  ->  u  =  ( U `  A ) )
119, 10, 8ifbieq12d 3785 . . . 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 2442 . . . 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 4374 . . . . 5  |-  ( ( t `  A )  i^i  ( U `  A ) )  e. 
_V
143, 13ifex 3821 . . . 4  |-  if ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/) ,  ( U `  A ) ,  ( ( t `
 A )  i^i  ( U `  A
) ) )  e. 
_V
157, 11, 12, 14ovmpt2 6238 . . 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 654 . 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 2474 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 set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727   _Vcvv 2962    i^i cin 3305   (/)c0 3613   ifcif 3763   U.cuni 4039   suc csuc 4612   omcom 4874   ran crn 4908   ` cfv 5483  (class class class)co 6110    e. cmpt2 6112  seq𝜔cseqom 6733
This theorem is referenced by:  fin23lem13  8243  fin23lem14  8244  fin23lem19  8247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-2nd 6379  df-recs 6662  df-rdg 6697  df-seqom 6734
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