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Theorem axdc3lem 8076
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 8073 . . . 4  |-  om  e.  _V
2 axdc3lem.1 . . . 4  |-  A  e. 
_V
31, 2xpex 4801 . . 3  |-  ( om 
X.  A )  e. 
_V
43pwex 4193 . 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 5400 . . . . . . . . 9  |-  ( s : suc  n --> A  -> 
s  C_  ( suc  n  X.  A ) )
7 peano2 4676 . . . . . . . . . 10  |-  ( n  e.  om  ->  suc  n  e.  om )
8 omelon2 4668 . . . . . . . . . . . 12  |-  ( om  e.  _V  ->  om  e.  On )
91, 8ax-mp 8 . . . . . . . . . . 11  |-  om  e.  On
109onelssi 4501 . . . . . . . . . 10  |-  ( suc  n  e.  om  ->  suc  n  C_  om )
11 xpss1 4795 . . . . . . . . . 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 3192 . . . . . . . 8  |-  ( ( s : suc  n --> A  /\  n  e.  om )  ->  s  C_  ( om  X.  A ) )
14 vex 2791 . . . . . . . . 9  |-  s  e. 
_V
1514elpw 3631 . . . . . . . 8  |-  ( s  e.  ~P ( om 
X.  A )  <->  s  C_  ( om  X.  A ) )
1613, 15sylibr 203 . . . . . . 7  |-  ( ( s : suc  n --> A  /\  n  e.  om )  ->  s  e.  ~P ( om  X.  A ) )
1716ancoms 439 . . . . . 6  |-  ( ( n  e.  om  /\  s : suc  n --> A )  ->  s  e.  ~P ( om  X.  A ) )
18173ad2antr1 1120 . . . . 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 ) )
1918rexlimiva 2662 . . . 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 ) )
2019abssi 3248 . . 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
)
215, 20eqsstri 3208 . 2  |-  S  C_  ~P ( om  X.  A
)
224, 21ssexi 4159 1  |-  S  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   Oncon0 4392   suc csuc 4394   omcom 4656    X. cxp 4687   -->wf 5251   ` cfv 5255
This theorem is referenced by:  axdc3lem2  8077  axdc3lem4  8079
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  ax-un 4512  ax-dc 8072
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-1o 6479
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