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Theorem fin23lem33 8256
Description: Lemma for fin23 8300. 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 5757 . . . . . . 7  |-  ( j  =  c  ->  (
e `  j )  =  ( e `  c ) )
21ineq1d 3527 . . . . . 6  |-  ( j  =  c  ->  (
( e `  j
)  i^i  k )  =  ( ( e `
 c )  i^i  k ) )
32eqeq1d 2450 . . . . 5  |-  ( j  =  c  ->  (
( ( e `  j )  i^i  k
)  =  (/)  <->  ( (
e `  c )  i^i  k )  =  (/) ) )
43, 2ifbieq2d 3783 . . . 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 3522 . . . . . 6  |-  ( k  =  d  ->  (
( e `  c
)  i^i  k )  =  ( ( e `
 c )  i^i  d ) )
65eqeq1d 2450 . . . . 5  |-  ( k  =  d  ->  (
( ( e `  c )  i^i  k
)  =  (/)  <->  ( (
e `  c )  i^i  d )  =  (/) ) )
7 id 21 . . . . 5  |-  ( k  =  d  ->  k  =  d )
86, 7, 5ifbieq12d 3785 . . . 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 6181 . . 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 2442 . . 3  |-  U. ran  e  =  U. ran  e
11 seqomeq12 6740 . . 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 655 . 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 5757 . . . 4  |-  ( l  =  y  ->  (
e `  l )  =  ( e `  y ) )
1514sseq2d 3362 . . 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 2961 . 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 2442 . 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 2442 . 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 2442 . 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 8255 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 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1727   {cab 2428   A.wral 2711   {crab 2715   _Vcvv 2962    \ cdif 3303    i^i cin 3305    C_ wss 3306    C. wpss 3307   (/)c0 3613   ifcif 3763   ~Pcpw 3823   U.cuni 4039   |^|cint 4074   class class class wbr 4237    e. cmpt 4291   suc csuc 4612   omcom 4874   ran crn 4908    o. ccom 4911   -1-1->wf1 5480   ` cfv 5483  (class class class)co 6110    e. cmpt2 6112   iota_crio 6571  seq𝜔cseqom 6733    ^m cmap 7047    ~~ cen 7135   Fincfn 7138
This theorem is referenced by:  fin23lem41  8263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-seqom 6734  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-card 7857
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