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Theorem fin23lem29 8622
Description: Lemma for fin23 8670. 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 3936 . . 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 5174 . . . . . 6  |-  ( Z  =  ( t  o.  R )  ->  ran  Z  =  ran  ( t  o.  R ) )
54unieqd 4210 . . . . 5  |-  ( Z  =  ( t  o.  R )  ->  U. ran  Z  =  U. ran  (
t  o.  R ) )
6 rncoss 5209 . . . . . 6  |-  ran  (
t  o.  R ) 
C_  ran  t
76unissi 4223 . . . . 5  |-  U. ran  ( t  o.  R
)  C_  U. ran  t
85, 7syl6eqss 3515 . . . 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 5174 . . . . . 6  |-  ( Z  =  ( ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  o.  Q )  ->  ran  Z  =  ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q ) )
1110unieqd 4210 . . . . 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 5209 . . . . . . 7  |-  ran  (
( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  o.  Q )  C_  ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )
1312unissi 4223 . . . . . 6  |-  U. ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q )  C_  U.
ran  ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )
14 unissb 4232 . . . . . . 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 2441 . . . . . . . . 9  |-  ( a  e.  { a  |  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) }  <->  E. z  e.  P  a  =  ( (
t `  z )  \  |^| ran  U ) )
16 fvssunirn 5823 . . . . . . . . . . . . 13  |-  ( t `
 z )  C_  U.
ran  t
1716a1i 11 . . . . . . . . . . . 12  |-  ( z  e.  P  ->  (
t `  z )  C_ 
U. ran  t )
1817ssdifssd 3603 . . . . . . . . . . 11  |-  ( z  e.  P  ->  (
( t `  z
)  \  |^| ran  U
)  C_  U. ran  t
)
19 sseq1 3486 . . . . . . . . . . 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 2941 . . . . . . . . 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 2454 . . . . . . . . 9  |-  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  =  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )
2423rnmpt 5194 . . . . . . . 8  |-  ran  (
z  e.  P  |->  ( ( t `  z
)  \  |^| ran  U
) )  =  {
a  |  E. z  e.  P  a  =  ( ( t `  z )  \  |^| ran 
U ) }
2522, 24eleq2s 2562 . . . . . . 7  |-  ( a  e.  ran  ( z  e.  P  |->  ( ( t `  z ) 
\  |^| ran  U ) )  ->  a  C_  U.
ran  t )
2614, 25mprgbir 2904 . . . . . 6  |-  U. ran  ( z  e.  P  |->  ( ( t `  z )  \  |^| ran 
U ) )  C_  U.
ran  t
2713, 26sstri 3474 . . . . 5  |-  U. ran  ( ( z  e.  P  |->  ( ( t `
 z )  \  |^| ran  U ) )  o.  Q )  C_  U.
ran  t
2811, 27syl6eqss 3515 . . . 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 1370    e. wcel 1758   {cab 2439   A.wral 2799   E.wrex 2800   {crab 2803   _Vcvv 3078    \ cdif 3434    i^i cin 3436    C_ wss 3437   (/)c0 3746   ifcif 3900   ~Pcpw 3969   U.cuni 4200   |^|cint 4237   class class class wbr 4401    |-> cmpt 4459   suc csuc 4830   ran crn 4950    o. ccom 4953   ` cfv 5527   iota_crio 6161  (class class class)co 6201    |-> cmpt2 6203   omcom 6587  seq𝜔cseqom 7013    ^m cmap 7325    ~~ cen 7418   Fincfn 7421
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-iota 5490  df-fv 5535
This theorem is referenced by:  fin23lem31  8624
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