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Theorem axdc3lem 8092
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 8089 . . . 4  |-  om  e.  _V
2 axdc3lem.1 . . . 4  |-  A  e. 
_V
31, 2xpex 4817 . . 3  |-  ( om 
X.  A )  e. 
_V
43pwex 4209 . 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 5416 . . . . . . . . 9  |-  ( s : suc  n --> A  -> 
s  C_  ( suc  n  X.  A ) )
7 peano2 4692 . . . . . . . . . 10  |-  ( n  e.  om  ->  suc  n  e.  om )
8 omelon2 4684 . . . . . . . . . . . 12  |-  ( om  e.  _V  ->  om  e.  On )
91, 8ax-mp 8 . . . . . . . . . . 11  |-  om  e.  On
109onelssi 4517 . . . . . . . . . 10  |-  ( suc  n  e.  om  ->  suc  n  C_  om )
11 xpss1 4811 . . . . . . . . . 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 3205 . . . . . . . 8  |-  ( ( s : suc  n --> A  /\  n  e.  om )  ->  s  C_  ( om  X.  A ) )
14 vex 2804 . . . . . . . . 9  |-  s  e. 
_V
1514elpw 3644 . . . . . . . 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 2675 . . . 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 3261 . . 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 3221 . 2  |-  S  C_  ~P ( om  X.  A
)
224, 21ssexi 4175 1  |-  S  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   Oncon0 4408   suc csuc 4410   omcom 4672    X. cxp 4703   -->wf 5267   ` cfv 5271
This theorem is referenced by:  axdc3lem2  8093  axdc3lem4  8095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-dc 8088
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-1o 6495
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