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Theorem fin23lem29 8221
Description: Lemma for fin23 8269. The residual is built from the same elements as the previous sequence. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem.a  |-  U  = seq𝜔 ( ( i  e.  om ,  u  e.  _V  |->  if ( ( ( t `
 i )  i^i  u )  =  (/) ,  u ,  ( ( t `  i )  i^i  u ) ) ) ,  U. ran  t )
fin23lem17.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 ) }
fin23lem.b  |-  P  =  { v  e.  om  |  |^| ran  U  C_  ( t `  v
) }
fin23lem.c  |-  Q  =  ( w  e.  om  |->  ( iota_ x  e.  P
( x  i^i  P
)  ~~  w )
)
fin23lem.d  |-  R  =  ( w  e.  om  |->  ( iota_ x  e.  ( om  \  P ) ( x  i^i  ( om  \  P ) ) 
~~  w ) )
fin23lem.e  |-  Z  =  if ( P  e. 
Fin ,  ( t  o.  R ) ,  ( ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  o.  Q ) )
Assertion
Ref Expression
fin23lem29  |-  U. ran  Z 
C_  U. ran  t
Distinct variable groups:    g, i,
t, u, v, x, z, a    F, a, t    w, a, x, z, P    v, a, R, i, u    U, a, i, u, v, z    Z, a    g, a
Allowed substitution hints:    P( v, u, t, g, i)    Q( x, z, w, v, u, t, g, i, a)    R( x, z, w, t, g)    U( x, w, t, g)    F( x, z, w, v, u, g, i)    Z( x, z, w, v, u, t, g, i)

Proof of Theorem fin23lem29
StepHypRef Expression
1 fin23lem.e . 2  |-  Z  =  if ( P  e. 
Fin ,  ( t  o.  R ) ,  ( ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  o.  Q ) )
2 eqif 3772 . . 3  |-  ( Z  =  if ( P  e.  Fin ,  ( t  o.  R ) ,  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q ) )  <->  ( ( P  e.  Fin  /\  Z  =  ( t  o.  R ) )  \/  ( -.  P  e. 
Fin  /\  Z  =  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q ) ) ) )
32biimpi 187 . 2  |-  ( Z  =  if ( P  e.  Fin ,  ( t  o.  R ) ,  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q ) )  ->  ( ( P  e.  Fin  /\  Z  =  ( t  o.  R ) )  \/  ( -.  P  e. 
Fin  /\  Z  =  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q ) ) ) )
4 rneq 5095 . . . . . 6  |-  ( Z  =  ( t  o.  R )  ->  ran  Z  =  ran  ( t  o.  R ) )
54unieqd 4026 . . . . 5  |-  ( Z  =  ( t  o.  R )  ->  U. ran  Z  =  U. ran  (
t  o.  R ) )
6 rncoss 5136 . . . . . 6  |-  ran  (
t  o.  R ) 
C_  ran  t
76unissi 4038 . . . . 5  |-  U. ran  ( t  o.  R
)  C_  U. ran  t
85, 7syl6eqss 3398 . . . 4  |-  ( Z  =  ( t  o.  R )  ->  U. ran  Z 
C_  U. ran  t )
98adantl 453 . . 3  |-  ( ( P  e.  Fin  /\  Z  =  ( t  o.  R ) )  ->  U. ran  Z  C_  U. ran  t )
10 rneq 5095 . . . . . 6  |-  ( Z  =  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q )  ->  ran  Z  =  ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q ) )
1110unieqd 4026 . . . . 5  |-  ( Z  =  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q )  ->  U. ran  Z  = 
U. ran  ( (
z  e.  P  |->  ( ( t `  z
)  \  |^| ran  U
) )  o.  Q
) )
12 rncoss 5136 . . . . . . 7  |-  ran  (
( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  o.  Q )  C_  ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )
1312unissi 4038 . . . . . 6  |-  U. ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q )  C_  U.
ran  ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )
14 unissb 4045 . . . . . . 7  |-  ( U. ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  C_  U.
ran  t  <->  A. a  e.  ran  ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) ) a  C_  U. ran  t
)
15 abid 2424 . . . . . . . . 9  |-  ( a  e.  { a  |  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) }  <->  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) )
16 fvssunirn 5754 . . . . . . . . . . . . 13  |-  ( t `
 z )  C_  U.
ran  t
1716a1i 11 . . . . . . . . . . . 12  |-  ( z  e.  P  ->  (
t `  z )  C_ 
U. ran  t )
1817ssdifssd 3485 . . . . . . . . . . 11  |-  ( z  e.  P  ->  (
( t `  z
)  \  |^| ran  U
)  C_  U. ran  t
)
19 sseq1 3369 . . . . . . . . . . 11  |-  ( a  =  ( ( t `
 z )  \  |^| ran  U )  -> 
( a  C_  U. ran  t 
<->  ( ( t `  z )  \  |^| ran 
U )  C_  U. ran  t ) )
2018, 19syl5ibrcom 214 . . . . . . . . . 10  |-  ( z  e.  P  ->  (
a  =  ( ( t `  z ) 
\  |^| ran  U )  ->  a  C_  U. ran  t ) )
2120rexlimiv 2824 . . . . . . . . 9  |-  ( E. z  e.  P  a  =  ( ( t `
 z )  \  |^| ran  U )  -> 
a  C_  U. ran  t
)
2215, 21sylbi 188 . . . . . . . 8  |-  ( a  e.  { a  |  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) }  ->  a  C_  U.
ran  t )
23 eqid 2436 . . . . . . . . 9  |-  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  =  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )
2423rnmpt 5116 . . . . . . . 8  |-  ran  (
z  e.  P  |->  ( ( t `  z
)  \  |^| ran  U
) )  =  {
a  |  E. z  e.  P  a  =  ( ( t `  z )  \  |^| ran 
U ) }
2522, 24eleq2s 2528 . . . . . . 7  |-  ( a  e.  ran  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  ->  a  C_  U.
ran  t )
2614, 25mprgbir 2776 . . . . . 6  |-  U. ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  C_  U.
ran  t
2713, 26sstri 3357 . . . . 5  |-  U. ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q )  C_  U.
ran  t
2811, 27syl6eqss 3398 . . . 4  |-  ( Z  =  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q )  ->  U. ran  Z  C_  U.
ran  t )
2928adantl 453 . . 3  |-  ( ( -.  P  e.  Fin  /\  Z  =  ( ( z  e.  P  |->  ( ( t `  z
)  \  |^| ran  U
) )  o.  Q
) )  ->  U. ran  Z 
C_  U. ran  t )
309, 29jaoi 369 . 2  |-  ( ( ( P  e.  Fin  /\  Z  =  ( t  o.  R ) )  \/  ( -.  P  e.  Fin  /\  Z  =  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q ) ) )  ->  U. ran  Z  C_ 
U. ran  t )
311, 3, 30mp2b 10 1  |-  U. ran  Z 
C_  U. ran  t
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   ifcif 3739   ~Pcpw 3799   U.cuni 4015   |^|cint 4050   class class class wbr 4212    e. cmpt 4266   suc csuc 4583   omcom 4845   ran crn 4879    o. ccom 4882   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   iota_crio 6542  seq𝜔cseqom 6704    ^m cmap 7018    ~~ cen 7106   Fincfn 7109
This theorem is referenced by:  fin23lem31  8223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fv 5462
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