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Theorem fin23lem12 7957
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 6469 . 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 5539 . . 3  |-  ( U `
 A )  e. 
_V
4 fveq2 5525 . . . . . . 7  |-  ( i  =  A  ->  (
t `  i )  =  ( t `  A ) )
54ineq1d 3369 . . . . . 6  |-  ( i  =  A  ->  (
( t `  i
)  i^i  u )  =  ( ( t `
 A )  i^i  u ) )
65eqeq1d 2291 . . . . 5  |-  ( i  =  A  ->  (
( ( t `  i )  i^i  u
)  =  (/)  <->  ( (
t `  A )  i^i  u )  =  (/) ) )
76, 5ifbieq2d 3585 . . . 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 3364 . . . . . 6  |-  ( u  =  ( U `  A )  ->  (
( t `  A
)  i^i  u )  =  ( ( t `
 A )  i^i  ( U `  A
) ) )
98eqeq1d 2291 . . . . 5  |-  ( u  =  ( U `  A )  ->  (
( ( t `  A )  i^i  u
)  =  (/)  <->  ( (
t `  A )  i^i  ( U `  A
) )  =  (/) ) )
10 id 19 . . . . 5  |-  ( u  =  ( U `  A )  ->  u  =  ( U `  A ) )
119, 10, 8ifbieq12d 3587 . . . 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 2283 . . . 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 4156 . . . . 5  |-  ( ( t `  A )  i^i  ( U `  A ) )  e. 
_V
143, 13ifex 3623 . . . 4  |-  if ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/) ,  ( U `  A ) ,  ( ( t `
 A )  i^i  ( U `  A
) ) )  e. 
_V
157, 11, 12, 14ovmpt2 5983 . . 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 652 . 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 2315 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 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   (/)c0 3455   ifcif 3565   U.cuni 3827   suc csuc 4394   omcom 4656   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860  seq𝜔cseqom 6459
This theorem is referenced by:  fin23lem13  7958  fin23lem14  7959  fin23lem19  7962
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-recs 6388  df-rdg 6423  df-seqom 6460
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