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Theorem fin23lem33 7971
Description: Lemma for fin23 8015. Discharge hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypothesis
Ref Expression
fin23lem33.f  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
Assertion
Ref Expression
fin23lem33  |-  ( G  e.  F  ->  E. f A. b ( ( b : om -1-1-> _V  /\  U.
ran  b  C_  G
)  ->  ( (
f `  b ) : om -1-1-> _V  /\  U. ran  ( f `  b
)  C.  U. ran  b
) ) )
Distinct variable groups:    a, b,
f, g, x, G    F, a
Allowed substitution hints:    F( x, f, g, b)

Proof of Theorem fin23lem33
Dummy variables  c 
d  e  i  j  k  l  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . . . 7  |-  ( j  =  c  ->  (
e `  j )  =  ( e `  c ) )
21ineq1d 3369 . . . . . 6  |-  ( j  =  c  ->  (
( e `  j
)  i^i  k )  =  ( ( e `
 c )  i^i  k ) )
32eqeq1d 2291 . . . . 5  |-  ( j  =  c  ->  (
( ( e `  j )  i^i  k
)  =  (/)  <->  ( (
e `  c )  i^i  k )  =  (/) ) )
43, 2ifbieq2d 3585 . . . 4  |-  ( j  =  c  ->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) )  =  if ( ( ( e `  c
)  i^i  k )  =  (/) ,  k ,  ( ( e `  c )  i^i  k
) ) )
5 ineq2 3364 . . . . . 6  |-  ( k  =  d  ->  (
( e `  c
)  i^i  k )  =  ( ( e `
 c )  i^i  d ) )
65eqeq1d 2291 . . . . 5  |-  ( k  =  d  ->  (
( ( e `  c )  i^i  k
)  =  (/)  <->  ( (
e `  c )  i^i  d )  =  (/) ) )
7 id 19 . . . . 5  |-  ( k  =  d  ->  k  =  d )
86, 7, 5ifbieq12d 3587 . . . 4  |-  ( k  =  d  ->  if ( ( ( e `
 c )  i^i  k )  =  (/) ,  k ,  ( ( e `  c )  i^i  k ) )  =  if ( ( ( e `  c
)  i^i  d )  =  (/) ,  d ,  ( ( e `  c )  i^i  d
) ) )
94, 8cbvmpt2v 5926 . . 3  |-  ( j  e.  om ,  k  e.  _V  |->  if ( ( ( e `  j )  i^i  k
)  =  (/) ,  k ,  ( ( e `
 j )  i^i  k ) ) )  =  ( c  e. 
om ,  d  e. 
_V  |->  if ( ( ( e `  c
)  i^i  d )  =  (/) ,  d ,  ( ( e `  c )  i^i  d
) ) )
10 eqid 2283 . . 3  |-  U. ran  e  =  U. ran  e
11 seqomeq12 6466 . . 3  |-  ( ( ( j  e.  om ,  k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) )  =  ( c  e.  om ,  d  e.  _V  |->  if ( ( ( e `  c )  i^i  d
)  =  (/) ,  d ,  ( ( e `
 c )  i^i  d ) ) )  /\  U. ran  e  =  U. ran  e )  -> seq𝜔
( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  = seq𝜔 ( ( c  e. 
om ,  d  e. 
_V  |->  if ( ( ( e `  c
)  i^i  d )  =  (/) ,  d ,  ( ( e `  c )  i^i  d
) ) ) , 
U. ran  e )
)
129, 10, 11mp2an 653 . 2  |- seq𝜔 ( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  = seq𝜔 ( ( c  e. 
om ,  d  e. 
_V  |->  if ( ( ( e `  c
)  i^i  d )  =  (/) ,  d ,  ( ( e `  c )  i^i  d
) ) ) , 
U. ran  e )
13 fin23lem33.f . 2  |-  F  =  { g  |  A. a  e.  ( ~P g  ^m  om ) ( A. x  e.  om  ( a `  suc  x )  C_  (
a `  x )  ->  |^| ran  a  e. 
ran  a ) }
14 fveq2 5525 . . . 4  |-  ( l  =  y  ->  (
e `  l )  =  ( e `  y ) )
1514sseq2d 3206 . . 3  |-  ( l  =  y  ->  ( |^| ran seq𝜔
( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  l
)  <->  |^| ran seq𝜔 ( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  y
) ) )
1615cbvrabv 2787 . 2  |-  { l  e.  om  |  |^| ran seq𝜔 (
( j  e.  om ,  k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  l ) }  =  { y  e.  om  |  |^| ran seq𝜔 ( ( j  e.  om , 
k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  y ) }
17 eqid 2283 . 2  |-  ( g  e.  om  |->  ( iota_ x  e.  { l  e. 
om  |  |^| ran seq𝜔 ( ( j  e.  om , 
k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  l ) }  ( x  i^i 
{ l  e.  om  |  |^| ran seq𝜔 ( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  l
) } )  ~~  g ) )  =  ( g  e.  om  |->  ( iota_ x  e.  {
l  e.  om  |  |^| ran seq𝜔
( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  l
) }  ( x  i^i  { l  e. 
om  |  |^| ran seq𝜔 ( ( j  e.  om , 
k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  l ) } )  ~~  g
) )
18 eqid 2283 . 2  |-  ( g  e.  om  |->  ( iota_ x  e.  ( om  \  {
l  e.  om  |  |^| ran seq𝜔
( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  l
) } ) ( x  i^i  ( om 
\  { l  e. 
om  |  |^| ran seq𝜔 ( ( j  e.  om , 
k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  l ) } ) )  ~~  g ) )  =  ( g  e.  om  |->  ( iota_ x  e.  ( om  \  { l  e.  om  |  |^| ran seq𝜔 (
( j  e.  om ,  k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  l ) } ) ( x  i^i  ( om  \  {
l  e.  om  |  |^| ran seq𝜔
( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  l
) } ) ) 
~~  g ) )
19 eqid 2283 . 2  |-  if ( { l  e.  om  |  |^| ran seq𝜔 ( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  l
) }  e.  Fin ,  ( e  o.  (
g  e.  om  |->  (
iota_ x  e.  ( om  \  { l  e. 
om  |  |^| ran seq𝜔 ( ( j  e.  om , 
k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  l ) } ) ( x  i^i  ( om  \  {
l  e.  om  |  |^| ran seq𝜔
( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  l
) } ) ) 
~~  g ) ) ) ,  ( ( i  e.  { l  e.  om  |  |^| ran seq𝜔 (
( j  e.  om ,  k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  l ) }  |->  ( ( e `
 i )  \  |^| ran seq𝜔
( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )
) )  o.  (
g  e.  om  |->  (
iota_ x  e.  { l  e.  om  |  |^| ran seq𝜔 (
( j  e.  om ,  k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  l ) }  ( x  i^i 
{ l  e.  om  |  |^| ran seq𝜔 ( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  l
) } )  ~~  g ) ) ) )  =  if ( { l  e.  om  |  |^| ran seq𝜔 ( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  l
) }  e.  Fin ,  ( e  o.  (
g  e.  om  |->  (
iota_ x  e.  ( om  \  { l  e. 
om  |  |^| ran seq𝜔 ( ( j  e.  om , 
k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  l ) } ) ( x  i^i  ( om  \  {
l  e.  om  |  |^| ran seq𝜔
( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  l
) } ) ) 
~~  g ) ) ) ,  ( ( i  e.  { l  e.  om  |  |^| ran seq𝜔 (
( j  e.  om ,  k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  l ) }  |->  ( ( e `
 i )  \  |^| ran seq𝜔
( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )
) )  o.  (
g  e.  om  |->  (
iota_ x  e.  { l  e.  om  |  |^| ran seq𝜔 (
( j  e.  om ,  k  e.  _V  |->  if ( ( ( e `
 j )  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k ) ) ) ,  U. ran  e )  C_  (
e `  l ) }  ( x  i^i 
{ l  e.  om  |  |^| ran seq𝜔 ( ( j  e. 
om ,  k  e. 
_V  |->  if ( ( ( e `  j
)  i^i  k )  =  (/) ,  k ,  ( ( e `  j )  i^i  k
) ) ) , 
U. ran  e )  C_  ( e `  l
) } )  ~~  g ) ) ) )
2012, 13, 16, 17, 18, 19fin23lem32 7970 1  |-  ( G  e.  F  ->  E. f A. b ( ( b : om -1-1-> _V  /\  U.
ran  b  C_  G
)  ->  ( (
f `  b ) : om -1-1-> _V  /\  U. ran  ( f `  b
)  C.  U. ran  b
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152    C. wpss 3153   (/)c0 3455   ifcif 3565   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   class class class wbr 4023    e. cmpt 4077   suc csuc 4394   omcom 4656   ran crn 4690    o. ccom 4693   -1-1->wf1 5252   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   iota_crio 6297  seq𝜔cseqom 6459    ^m cmap 6772    ~~ cen 6860   Fincfn 6863
This theorem is referenced by:  fin23lem41  7978
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-rep 4131  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-rmo 2551  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-int 3863  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-seqom 6460  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572
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