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Theorem fin23lem12 8175
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 6681 . 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 5709 . . 3  |-  ( U `
 A )  e. 
_V
4 fveq2 5695 . . . . . . 7  |-  ( i  =  A  ->  (
t `  i )  =  ( t `  A ) )
54ineq1d 3509 . . . . . 6  |-  ( i  =  A  ->  (
( t `  i
)  i^i  u )  =  ( ( t `
 A )  i^i  u ) )
65eqeq1d 2420 . . . . 5  |-  ( i  =  A  ->  (
( ( t `  i )  i^i  u
)  =  (/)  <->  ( (
t `  A )  i^i  u )  =  (/) ) )
76, 5ifbieq2d 3727 . . . 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 3504 . . . . . 6  |-  ( u  =  ( U `  A )  ->  (
( t `  A
)  i^i  u )  =  ( ( t `
 A )  i^i  ( U `  A
) ) )
98eqeq1d 2420 . . . . 5  |-  ( u  =  ( U `  A )  ->  (
( ( t `  A )  i^i  u
)  =  (/)  <->  ( (
t `  A )  i^i  ( U `  A
) )  =  (/) ) )
10 id 20 . . . . 5  |-  ( u  =  ( U `  A )  ->  u  =  ( U `  A ) )
119, 10, 8ifbieq12d 3729 . . . 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 2412 . . . 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 4313 . . . . 5  |-  ( ( t `  A )  i^i  ( U `  A ) )  e. 
_V
143, 13ifex 3765 . . . 4  |-  if ( ( ( t `  A )  i^i  ( U `  A )
)  =  (/) ,  ( U `  A ) ,  ( ( t `
 A )  i^i  ( U `  A
) ) )  e. 
_V
157, 11, 12, 14ovmpt2 6176 . . 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 653 . 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 2444 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 1649    e. wcel 1721   _Vcvv 2924    i^i cin 3287   (/)c0 3596   ifcif 3707   U.cuni 3983   suc csuc 4551   omcom 4812   ran crn 4846   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050  seq𝜔cseqom 6671
This theorem is referenced by:  fin23lem13  8176  fin23lem14  8177  fin23lem19  8180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-2nd 6317  df-recs 6600  df-rdg 6635  df-seqom 6672
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