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Theorem fin23lem33 8716
Description: Lemma for fin23 8760. 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 5848 . . . . . . 7  |-  ( j  =  c  ->  (
e `  j )  =  ( e `  c ) )
21ineq1d 3685 . . . . . 6  |-  ( j  =  c  ->  (
( e `  j
)  i^i  k )  =  ( ( e `
 c )  i^i  k ) )
32eqeq1d 2456 . . . . 5  |-  ( j  =  c  ->  (
( ( e `  j )  i^i  k
)  =  (/)  <->  ( (
e `  c )  i^i  k )  =  (/) ) )
43, 2ifbieq2d 3954 . . . 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 3680 . . . . . 6  |-  ( k  =  d  ->  (
( e `  c
)  i^i  k )  =  ( ( e `
 c )  i^i  d ) )
65eqeq1d 2456 . . . . 5  |-  ( k  =  d  ->  (
( ( e `  c )  i^i  k
)  =  (/)  <->  ( (
e `  c )  i^i  d )  =  (/) ) )
7 id 22 . . . . 5  |-  ( k  =  d  ->  k  =  d )
86, 7, 5ifbieq12d 3956 . . . 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 6350 . . 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 2454 . . 3  |-  U. ran  e  =  U. ran  e
11 seqomeq12 7111 . . 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 670 . 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 5848 . . . 4  |-  ( l  =  y  ->  (
e `  l )  =  ( e `  y ) )
1514sseq2d 3517 . . 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 3105 . 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 2454 . 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 2454 . 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 2454 . 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 8715 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823   {cab 2439   A.wral 2804   {crab 2808   _Vcvv 3106    \ cdif 3458    i^i cin 3460    C_ wss 3461    C. wpss 3462   (/)c0 3783   ifcif 3929   ~Pcpw 3999   U.cuni 4235   |^|cint 4271   class class class wbr 4439    |-> cmpt 4497   suc csuc 4869   ran crn 4989    o. ccom 4992   -1-1->wf1 5567   ` cfv 5570   iota_crio 6231  (class class class)co 6270    |-> cmpt2 6272   omcom 6673  seq𝜔cseqom 7104    ^m cmap 7412    ~~ cen 7506   Fincfn 7509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311
This theorem is referenced by:  fin23lem41  8723
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