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Theorem axdc3lem 8878
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 8875 . . . 4  |-  om  e.  _V
2 axdc3lem.1 . . . 4  |-  A  e. 
_V
31, 2xpex 6609 . . 3  |-  ( om 
X.  A )  e. 
_V
43pwex 4608 . 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 5758 . . . . . . . . 9  |-  ( s : suc  n --> A  -> 
s  C_  ( suc  n  X.  A ) )
7 peano2 6727 . . . . . . . . . 10  |-  ( n  e.  om  ->  suc  n  e.  om )
8 omelon2 6718 . . . . . . . . . . . 12  |-  ( om  e.  _V  ->  om  e.  On )
91, 8ax-mp 5 . . . . . . . . . . 11  |-  om  e.  On
109onelssi 5550 . . . . . . . . . 10  |-  ( suc  n  e.  om  ->  suc  n  C_  om )
11 xpss1 4963 . . . . . . . . . 10  |-  ( suc  n  C_  om  ->  ( suc  n  X.  A
)  C_  ( om  X.  A ) )
127, 10, 113syl 18 . . . . . . . . 9  |-  ( n  e.  om  ->  ( suc  n  X.  A ) 
C_  ( om  X.  A ) )
136, 12sylan9ss 3483 . . . . . . . 8  |-  ( ( s : suc  n --> A  /\  n  e.  om )  ->  s  C_  ( om  X.  A ) )
14 selpw 3992 . . . . . . . 8  |-  ( s  e.  ~P ( om 
X.  A )  <->  s  C_  ( om  X.  A ) )
1513, 14sylibr 215 . . . . . . 7  |-  ( ( s : suc  n --> A  /\  n  e.  om )  ->  s  e.  ~P ( om  X.  A ) )
1615ancoms 454 . . . . . 6  |-  ( ( n  e.  om  /\  s : suc  n --> A )  ->  s  e.  ~P ( om  X.  A ) )
17163ad2antr1 1170 . . . . 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 2920 . . . 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 3542 . . 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 3500 . 2  |-  S  C_  ~P ( om  X.  A
)
214, 20ssexi 4570 1  |-  S  e. 
_V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {cab 2414   A.wral 2782   E.wrex 2783   _Vcvv 3087    C_ wss 3442   (/)c0 3767   ~Pcpw 3985    X. cxp 4852   Oncon0 5442   suc csuc 5444   -->wf 5597   ` cfv 5601   omcom 6706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-dc 8874
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-om 6707  df-1o 7190
This theorem is referenced by:  axdc3lem2  8879  axdc3lem4  8881
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