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Theorem fin23lem33 8061
Description: Lemma for fin23 8105. 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 5608 . . . . . . 7  |-  ( j  =  c  ->  (
e `  j )  =  ( e `  c ) )
21ineq1d 3445 . . . . . 6  |-  ( j  =  c  ->  (
( e `  j
)  i^i  k )  =  ( ( e `
 c )  i^i  k ) )
32eqeq1d 2366 . . . . 5  |-  ( j  =  c  ->  (
( ( e `  j )  i^i  k
)  =  (/)  <->  ( (
e `  c )  i^i  k )  =  (/) ) )
43, 2ifbieq2d 3661 . . . 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 3440 . . . . . 6  |-  ( k  =  d  ->  (
( e `  c
)  i^i  k )  =  ( ( e `
 c )  i^i  d ) )
65eqeq1d 2366 . . . . 5  |-  ( k  =  d  ->  (
( ( e `  c )  i^i  k
)  =  (/)  <->  ( (
e `  c )  i^i  d )  =  (/) ) )
7 id 19 . . . . 5  |-  ( k  =  d  ->  k  =  d )
86, 7, 5ifbieq12d 3663 . . . 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 6013 . . 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 2358 . . 3  |-  U. ran  e  =  U. ran  e
11 seqomeq12 6553 . . 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 5608 . . . 4  |-  ( l  =  y  ->  (
e `  l )  =  ( e `  y ) )
1514sseq2d 3282 . . 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 2863 . 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 2358 . 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 2358 . 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 2358 . 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 8060 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 1540   E.wex 1541    = wceq 1642    e. wcel 1710   {cab 2344   A.wral 2619   {crab 2623   _Vcvv 2864    \ cdif 3225    i^i cin 3227    C_ wss 3228    C. wpss 3229   (/)c0 3531   ifcif 3641   ~Pcpw 3701   U.cuni 3908   |^|cint 3943   class class class wbr 4104    e. cmpt 4158   suc csuc 4476   omcom 4738   ran crn 4772    o. ccom 4775   -1-1->wf1 5334   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   iota_crio 6384  seq𝜔cseqom 6546    ^m cmap 6860    ~~ cen 6948   Fincfn 6951
This theorem is referenced by:  fin23lem41  8068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-seqom 6547  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-card 7662
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