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Theorem axdc3lem 8829
Description: The class  S of finite approximations to the DC sequence is a set. (We derive here the stronger statement that  S is a subset of a specific set, namely  ~P ( om  X.  A ).) (Unnecessary distinct variable restrictions were removed by David Abernethy, 18-Mar-2014.) (Contributed by Mario Carneiro, 27-Jan-2013.) (Revised by Mario Carneiro, 18-Mar-2014.)
Hypotheses
Ref Expression
axdc3lem.1  |-  A  e. 
_V
axdc3lem.2  |-  S  =  { s  |  E. n  e.  om  (
s : suc  n --> A  /\  ( s `  (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k
)  e.  ( F `
 ( s `  k ) ) ) }
Assertion
Ref Expression
axdc3lem  |-  S  e. 
_V
Distinct variable group:    A, n, s
Allowed substitution hints:    A( k)    C( k, n, s)    S( k, n, s)    F( k, n, s)

Proof of Theorem axdc3lem
StepHypRef Expression
1 dcomex 8826 . . . 4  |-  om  e.  _V
2 axdc3lem.1 . . . 4  |-  A  e. 
_V
31, 2xpex 6712 . . 3  |-  ( om 
X.  A )  e. 
_V
43pwex 4630 . 2  |-  ~P ( om  X.  A )  e. 
_V
5 axdc3lem.2 . . 3  |-  S  =  { s  |  E. n  e.  om  (
s : suc  n --> A  /\  ( s `  (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k
)  e.  ( F `
 ( s `  k ) ) ) }
6 fssxp 5742 . . . . . . . . 9  |-  ( s : suc  n --> A  -> 
s  C_  ( suc  n  X.  A ) )
7 peano2 6699 . . . . . . . . . 10  |-  ( n  e.  om  ->  suc  n  e.  om )
8 omelon2 6691 . . . . . . . . . . . 12  |-  ( om  e.  _V  ->  om  e.  On )
91, 8ax-mp 5 . . . . . . . . . . 11  |-  om  e.  On
109onelssi 4986 . . . . . . . . . 10  |-  ( suc  n  e.  om  ->  suc  n  C_  om )
11 xpss1 5110 . . . . . . . . . 10  |-  ( suc  n  C_  om  ->  ( suc  n  X.  A
)  C_  ( om  X.  A ) )
127, 10, 113syl 20 . . . . . . . . 9  |-  ( n  e.  om  ->  ( suc  n  X.  A ) 
C_  ( om  X.  A ) )
136, 12sylan9ss 3517 . . . . . . . 8  |-  ( ( s : suc  n --> A  /\  n  e.  om )  ->  s  C_  ( om  X.  A ) )
14 selpw 4017 . . . . . . . 8  |-  ( s  e.  ~P ( om 
X.  A )  <->  s  C_  ( om  X.  A ) )
1513, 14sylibr 212 . . . . . . 7  |-  ( ( s : suc  n --> A  /\  n  e.  om )  ->  s  e.  ~P ( om  X.  A ) )
1615ancoms 453 . . . . . 6  |-  ( ( n  e.  om  /\  s : suc  n --> A )  ->  s  e.  ~P ( om  X.  A ) )
17163ad2antr1 1161 . . . . 5  |-  ( ( n  e.  om  /\  ( s : suc  n
--> A  /\  ( s `
 (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k )  e.  ( F `  ( s `
 k ) ) ) )  ->  s  e.  ~P ( om  X.  A ) )
1817rexlimiva 2951 . . . 4  |-  ( E. n  e.  om  (
s : suc  n --> A  /\  ( s `  (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k
)  e.  ( F `
 ( s `  k ) ) )  ->  s  e.  ~P ( om  X.  A ) )
1918abssi 3575 . . 3  |-  { s  |  E. n  e. 
om  ( s : suc  n --> A  /\  ( s `  (/) )  =  C  /\  A. k  e.  n  ( s `  suc  k )  e.  ( F `  (
s `  k )
) ) }  C_  ~P ( om  X.  A
)
205, 19eqsstri 3534 . 2  |-  S  C_  ~P ( om  X.  A
)
214, 20ssexi 4592 1  |-  S  e. 
_V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   Oncon0 4878   suc csuc 4880    X. cxp 4997   -->wf 5583   ` cfv 5587   omcom 6679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-dc 8825
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-fv 5595  df-om 6680  df-1o 7130
This theorem is referenced by:  axdc3lem2  8830  axdc3lem4  8832
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