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Theorem axdc3lem 8330
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 8327 . . . 4  |-  om  e.  _V
2 axdc3lem.1 . . . 4  |-  A  e. 
_V
31, 2xpex 4990 . . 3  |-  ( om 
X.  A )  e. 
_V
43pwex 4382 . 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 5602 . . . . . . . . 9  |-  ( s : suc  n --> A  -> 
s  C_  ( suc  n  X.  A ) )
7 peano2 4865 . . . . . . . . . 10  |-  ( n  e.  om  ->  suc  n  e.  om )
8 omelon2 4857 . . . . . . . . . . . 12  |-  ( om  e.  _V  ->  om  e.  On )
91, 8ax-mp 8 . . . . . . . . . . 11  |-  om  e.  On
109onelssi 4690 . . . . . . . . . 10  |-  ( suc  n  e.  om  ->  suc  n  C_  om )
11 xpss1 4984 . . . . . . . . . 10  |-  ( suc  n  C_  om  ->  ( suc  n  X.  A
)  C_  ( om  X.  A ) )
127, 10, 113syl 19 . . . . . . . . 9  |-  ( n  e.  om  ->  ( suc  n  X.  A ) 
C_  ( om  X.  A ) )
136, 12sylan9ss 3361 . . . . . . . 8  |-  ( ( s : suc  n --> A  /\  n  e.  om )  ->  s  C_  ( om  X.  A ) )
14 vex 2959 . . . . . . . . 9  |-  s  e. 
_V
1514elpw 3805 . . . . . . . 8  |-  ( s  e.  ~P ( om 
X.  A )  <->  s  C_  ( om  X.  A ) )
1613, 15sylibr 204 . . . . . . 7  |-  ( ( s : suc  n --> A  /\  n  e.  om )  ->  s  e.  ~P ( om  X.  A ) )
1716ancoms 440 . . . . . 6  |-  ( ( n  e.  om  /\  s : suc  n --> A )  ->  s  e.  ~P ( om  X.  A ) )
18173ad2antr1 1122 . . . . 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 2825 . . . 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 3418 . . 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 3378 . 2  |-  S  C_  ~P ( om  X.  A
)
224, 21ssexi 4348 1  |-  S  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   E.wrex 2706   _Vcvv 2956    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   Oncon0 4581   suc csuc 4583   omcom 4845    X. cxp 4876   -->wf 5450   ` cfv 5454
This theorem is referenced by:  axdc3lem2  8331  axdc3lem4  8333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-dc 8326
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-1o 6724
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