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Theorem fin23lem12 8521
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 6933 . 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 5722 . . 3  |-  ( U `
 A )  e. 
_V
4 fveq2 5712 . . . . . . 7  |-  ( i  =  A  ->  (
t `  i )  =  ( t `  A ) )
54ineq1d 3572 . . . . . 6  |-  ( i  =  A  ->  (
( t `  i
)  i^i  u )  =  ( ( t `
 A )  i^i  u ) )
65eqeq1d 2451 . . . . 5  |-  ( i  =  A  ->  (
( ( t `  i )  i^i  u
)  =  (/)  <->  ( (
t `  A )  i^i  u )  =  (/) ) )
76, 5ifbieq2d 3835 . . . 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 3567 . . . . . 6  |-  ( u  =  ( U `  A )  ->  (
( t `  A
)  i^i  u )  =  ( ( t `
 A )  i^i  ( U `  A
) ) )
98eqeq1d 2451 . . . . 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 3837 . . . 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 2443 . . . 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 4455 . . . . 5  |-  ( ( t `  A )  i^i  ( U `  A ) )  e. 
_V
143, 13ifex 3879 . . . 4  |-  if ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/) ,  ( U `  A ) ,  ( ( t `
 A )  i^i  ( U `  A
) ) )  e. 
_V
157, 11, 12, 14ovmpt2 6247 . . 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 2475 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 1369    e. wcel 1756   _Vcvv 2993    i^i cin 3348   (/)c0 3658   ifcif 3812   U.cuni 4112   suc csuc 4742   ran crn 4862   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   omcom 6497  seq𝜔cseqom 6923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-2nd 6599  df-recs 6853  df-rdg 6887  df-seqom 6924
This theorem is referenced by:  fin23lem13  8522  fin23lem14  8523  fin23lem19  8526
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