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Theorem fin23lem29 8738
Description: Lemma for fin23 8786. 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 3982 . . 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 194 . 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 5238 . . . . . 6  |-  ( Z  =  ( t  o.  R )  ->  ran  Z  =  ran  ( t  o.  R ) )
54unieqd 4261 . . . . 5  |-  ( Z  =  ( t  o.  R )  ->  U. ran  Z  =  U. ran  (
t  o.  R ) )
6 rncoss 5273 . . . . . 6  |-  ran  (
t  o.  R ) 
C_  ran  t
76unissi 4274 . . . . 5  |-  U. ran  ( t  o.  R
)  C_  U. ran  t
85, 7syl6eqss 3549 . . . 4  |-  ( Z  =  ( t  o.  R )  ->  U. ran  Z 
C_  U. ran  t )
98adantl 466 . . 3  |-  ( ( P  e.  Fin  /\  Z  =  ( t  o.  R ) )  ->  U. ran  Z  C_  U. ran  t )
10 rneq 5238 . . . . . 6  |-  ( Z  =  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q )  ->  ran  Z  =  ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q ) )
1110unieqd 4261 . . . . 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 5273 . . . . . . 7  |-  ran  (
( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  o.  Q )  C_  ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )
1312unissi 4274 . . . . . 6  |-  U. ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q )  C_  U.
ran  ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )
14 unissb 4283 . . . . . . 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 2444 . . . . . . . . 9  |-  ( a  e.  { a  |  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) }  <->  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) )
16 fvssunirn 5895 . . . . . . . . . . . . 13  |-  ( t `
 z )  C_  U.
ran  t
1716a1i 11 . . . . . . . . . . . 12  |-  ( z  e.  P  ->  (
t `  z )  C_ 
U. ran  t )
1817ssdifssd 3638 . . . . . . . . . . 11  |-  ( z  e.  P  ->  (
( t `  z
)  \  |^| ran  U
)  C_  U. ran  t
)
19 sseq1 3520 . . . . . . . . . . 11  |-  ( a  =  ( ( t `
 z )  \  |^| ran  U )  -> 
( a  C_  U. ran  t 
<->  ( ( t `  z )  \  |^| ran 
U )  C_  U. ran  t ) )
2018, 19syl5ibrcom 222 . . . . . . . . . 10  |-  ( z  e.  P  ->  (
a  =  ( ( t `  z ) 
\  |^| ran  U )  ->  a  C_  U. ran  t ) )
2120rexlimiv 2943 . . . . . . . . 9  |-  ( E. z  e.  P  a  =  ( ( t `
 z )  \  |^| ran  U )  -> 
a  C_  U. ran  t
)
2215, 21sylbi 195 . . . . . . . 8  |-  ( a  e.  { a  |  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) }  ->  a  C_  U.
ran  t )
23 eqid 2457 . . . . . . . . 9  |-  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  =  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )
2423rnmpt 5258 . . . . . . . 8  |-  ran  (
z  e.  P  |->  ( ( t `  z
)  \  |^| ran  U
) )  =  {
a  |  E. z  e.  P  a  =  ( ( t `  z )  \  |^| ran 
U ) }
2522, 24eleq2s 2565 . . . . . . 7  |-  ( a  e.  ran  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  ->  a  C_  U.
ran  t )
2614, 25mprgbir 2821 . . . . . 6  |-  U. ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  C_  U.
ran  t
2713, 26sstri 3508 . . . . 5  |-  U. ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q )  C_  U.
ran  t
2811, 27syl6eqss 3549 . . . 4  |-  ( Z  =  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q )  ->  U. ran  Z  C_  U.
ran  t )
2928adantl 466 . . 3  |-  ( ( -.  P  e.  Fin  /\  Z  =  ( ( z  e.  P  |->  ( ( t `  z
)  \  |^| ran  U
) )  o.  Q
) )  ->  U. ran  Z 
C_  U. ran  t )
309, 29jaoi 379 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109    \ cdif 3468    i^i cin 3470    C_ wss 3471   (/)c0 3793   ifcif 3944   ~Pcpw 4015   U.cuni 4251   |^|cint 4288   class class class wbr 4456    |-> cmpt 4515   suc csuc 4889   ran crn 5009    o. ccom 5012   ` cfv 5594   iota_crio 6257  (class class class)co 6296    |-> cmpt2 6298   omcom 6699  seq𝜔cseqom 7130    ^m cmap 7438    ~~ cen 7532   Fincfn 7535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fv 5602
This theorem is referenced by:  fin23lem31  8740
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