MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem33 Structured version   Unicode version

Theorem fin23lem33 8617
Description: Lemma for fin23 8661. 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 5791 . . . . . . 7  |-  ( j  =  c  ->  (
e `  j )  =  ( e `  c ) )
21ineq1d 3651 . . . . . 6  |-  ( j  =  c  ->  (
( e `  j
)  i^i  k )  =  ( ( e `
 c )  i^i  k ) )
32eqeq1d 2453 . . . . 5  |-  ( j  =  c  ->  (
( ( e `  j )  i^i  k
)  =  (/)  <->  ( (
e `  c )  i^i  k )  =  (/) ) )
43, 2ifbieq2d 3914 . . . 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 3646 . . . . . 6  |-  ( k  =  d  ->  (
( e `  c
)  i^i  k )  =  ( ( e `
 c )  i^i  d ) )
65eqeq1d 2453 . . . . 5  |-  ( k  =  d  ->  (
( ( e `  c )  i^i  k
)  =  (/)  <->  ( (
e `  c )  i^i  d )  =  (/) ) )
7 id 22 . . . . 5  |-  ( k  =  d  ->  k  =  d )
86, 7, 5ifbieq12d 3916 . . . 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 6267 . . 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 2451 . . 3  |-  U. ran  e  =  U. ran  e
11 seqomeq12 7011 . . 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 672 . 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 5791 . . . 4  |-  ( l  =  y  ->  (
e `  l )  =  ( e `  y ) )
1514sseq2d 3484 . . 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 3069 . 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 2451 . 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 2451 . 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 2451 . 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 8616 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 369   A.wal 1368    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2436   A.wral 2795   {crab 2799   _Vcvv 3070    \ cdif 3425    i^i cin 3427    C_ wss 3428    C. wpss 3429   (/)c0 3737   ifcif 3891   ~Pcpw 3960   U.cuni 4191   |^|cint 4228   class class class wbr 4392    |-> cmpt 4450   suc csuc 4821   ran crn 4941    o. ccom 4944   -1-1->wf1 5515   ` cfv 5518   iota_crio 6152  (class class class)co 6192    |-> cmpt2 6194   omcom 6578  seq𝜔cseqom 7004    ^m cmap 7316    ~~ cen 7409   Fincfn 7412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-seqom 7005  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212
This theorem is referenced by:  fin23lem41  8624
  Copyright terms: Public domain W3C validator