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Theorem fin23lem29 7983
Description: Lemma for fin23 8031. 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 3611 . . 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 4961 . . . . . 6  |-  ran  (
t  o.  R ) 
C_  ran  t
5 uniss 3864 . . . . . 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 4920 . . . . . . 7  |-  ( Z  =  ( t  o.  R )  ->  ran  Z  =  ran  ( t  o.  R ) )
87unieqd 3854 . . . . . 6  |-  ( Z  =  ( t  o.  R )  ->  U. ran  Z  =  U. ran  (
t  o.  R ) )
98sseq1d 3218 . . . . 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 4961 . . . . . . 7  |-  ran  (
( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  o.  Q )  C_  ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )
13 uniss 3864 . . . . . . 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 3873 . . . . . . 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 2284 . . . . . . . . 9  |-  ( a  e.  { a  |  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) }  <->  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) )
17 difss 3316 . . . . . . . . . . . 12  |-  ( ( t `  z ) 
\  |^| ran  U ) 
C_  ( t `  z )
18 fvssunirn 5567 . . . . . . . . . . . . 13  |-  ( t `
 z )  C_  U.
ran  t
1918a1i 10 . . . . . . . . . . . 12  |-  ( z  e.  P  ->  (
t `  z )  C_ 
U. ran  t )
2017, 19syl5ss 3203 . . . . . . . . . . 11  |-  ( z  e.  P  ->  (
( t `  z
)  \  |^| ran  U
)  C_  U. ran  t
)
21 sseq1 3212 . . . . . . . . . . 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 2674 . . . . . . . . 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 2296 . . . . . . . . 9  |-  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  =  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )
2625rnmpt 4941 . . . . . . . 8  |-  ran  (
z  e.  P  |->  ( ( t `  z
)  \  |^| ran  U
) )  =  {
a  |  E. z  e.  P  a  =  ( ( t `  z )  \  |^| ran 
U ) }
2724, 26eleq2s 2388 . . . . . . 7  |-  ( a  e.  ran  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  ->  a  C_  U.
ran  t )
2815, 27mprgbir 2626 . . . . . 6  |-  U. ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  C_  U.
ran  t
2914, 28sstri 3201 . . . . 5  |-  U. ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q )  C_  U.
ran  t
30 rneq 4920 . . . . . . 7  |-  ( Z  =  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q )  ->  ran  Z  =  ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q ) )
3130unieqd 3854 . . . . . 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 3218 . . . . 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 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638   U.cuni 3843   |^|cint 3878   class class class wbr 4039    e. cmpt 4093   suc csuc 4410   omcom 4672   ran crn 4706    o. ccom 4709   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   iota_crio 6313  seq𝜔cseqom 6475    ^m cmap 6788    ~~ cen 6876   Fincfn 6879
This theorem is referenced by:  fin23lem31  7985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fv 5279
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