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Theorem fin23lem29 7967
Description: Lemma for fin23 8015. 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 3598 . . 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 186 . 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 rncoss 4945 . . . . . 6  |-  ran  (
t  o.  R ) 
C_  ran  t
5 uniss 3848 . . . . . 6  |-  ( ran  ( t  o.  R
)  C_  ran  t  ->  U. ran  ( t  o.  R )  C_  U. ran  t )
64, 5ax-mp 8 . . . . 5  |-  U. ran  ( t  o.  R
)  C_  U. ran  t
7 rneq 4904 . . . . . . 7  |-  ( Z  =  ( t  o.  R )  ->  ran  Z  =  ran  ( t  o.  R ) )
87unieqd 3838 . . . . . 6  |-  ( Z  =  ( t  o.  R )  ->  U. ran  Z  =  U. ran  (
t  o.  R ) )
98sseq1d 3205 . . . . 5  |-  ( Z  =  ( t  o.  R )  ->  ( U. ran  Z  C_  U. ran  t 
<-> 
U. ran  ( t  o.  R )  C_  U. ran  t ) )
106, 9mpbiri 224 . . . 4  |-  ( Z  =  ( t  o.  R )  ->  U. ran  Z 
C_  U. ran  t )
1110adantl 452 . . 3  |-  ( ( P  e.  Fin  /\  Z  =  ( t  o.  R ) )  ->  U. ran  Z  C_  U. ran  t )
12 rncoss 4945 . . . . . . 7  |-  ran  (
( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  o.  Q )  C_  ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )
13 uniss 3848 . . . . . . 7  |-  ( ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q )  C_  ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  ->  U. ran  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q ) 
C_  U. ran  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) ) )
1412, 13ax-mp 8 . . . . . 6  |-  U. ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q )  C_  U.
ran  ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )
15 unissb 3857 . . . . . . 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
)
16 abid 2271 . . . . . . . . 9  |-  ( a  e.  { a  |  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) }  <->  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) )
17 difss 3303 . . . . . . . . . . . 12  |-  ( ( t `  z ) 
\  |^| ran  U ) 
C_  ( t `  z )
18 fvssunirn 5551 . . . . . . . . . . . . 13  |-  ( t `
 z )  C_  U.
ran  t
1918a1i 10 . . . . . . . . . . . 12  |-  ( z  e.  P  ->  (
t `  z )  C_ 
U. ran  t )
2017, 19syl5ss 3190 . . . . . . . . . . 11  |-  ( z  e.  P  ->  (
( t `  z
)  \  |^| ran  U
)  C_  U. ran  t
)
21 sseq1 3199 . . . . . . . . . . 11  |-  ( a  =  ( ( t `
 z )  \  |^| ran  U )  -> 
( a  C_  U. ran  t 
<->  ( ( t `  z )  \  |^| ran 
U )  C_  U. ran  t ) )
2220, 21syl5ibrcom 213 . . . . . . . . . 10  |-  ( z  e.  P  ->  (
a  =  ( ( t `  z ) 
\  |^| ran  U )  ->  a  C_  U. ran  t ) )
2322rexlimiv 2661 . . . . . . . . 9  |-  ( E. z  e.  P  a  =  ( ( t `
 z )  \  |^| ran  U )  -> 
a  C_  U. ran  t
)
2416, 23sylbi 187 . . . . . . . 8  |-  ( a  e.  { a  |  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) }  ->  a  C_  U.
ran  t )
25 eqid 2283 . . . . . . . . 9  |-  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  =  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )
2625rnmpt 4925 . . . . . . . 8  |-  ran  (
z  e.  P  |->  ( ( t `  z
)  \  |^| ran  U
) )  =  {
a  |  E. z  e.  P  a  =  ( ( t `  z )  \  |^| ran 
U ) }
2724, 26eleq2s 2375 . . . . . . 7  |-  ( a  e.  ran  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  ->  a  C_  U.
ran  t )
2815, 27mprgbir 2613 . . . . . 6  |-  U. ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  C_  U.
ran  t
2914, 28sstri 3188 . . . . 5  |-  U. ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q )  C_  U.
ran  t
30 rneq 4904 . . . . . . 7  |-  ( Z  =  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q )  ->  ran  Z  =  ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q ) )
3130unieqd 3838 . . . . . 6  |-  ( Z  =  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q )  ->  U. ran  Z  = 
U. ran  ( (
z  e.  P  |->  ( ( t `  z
)  \  |^| ran  U
) )  o.  Q
) )
3231sseq1d 3205 . . . . 5  |-  ( Z  =  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q )  ->  ( U. ran  Z 
C_  U. ran  t  <->  U. ran  (
( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  o.  Q )  C_  U. ran  t ) )
3329, 32mpbiri 224 . . . 4  |-  ( Z  =  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q )  ->  U. ran  Z  C_  U.
ran  t )
3433adantl 452 . . 3  |-  ( ( -.  P  e.  Fin  /\  Z  =  ( ( z  e.  P  |->  ( ( t `  z
)  \  |^| ran  U
) )  o.  Q
) )  ->  U. ran  Z 
C_  U. ran  t )
3511, 34jaoi 368 . 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 )
361, 3, 35mp2b 9 1  |-  U. ran  Z 
C_  U. ran  t
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   class class class wbr 4023    e. cmpt 4077   suc csuc 4394   omcom 4656   ran crn 4690    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   iota_crio 6297  seq𝜔cseqom 6459    ^m cmap 6772    ~~ cen 6860   Fincfn 6863
This theorem is referenced by:  fin23lem31  7969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fv 5263
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