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Theorem vtarsuelt 25307
Description: C belongs to the value of  tar at a successor of  A iff it is a part of  tar at  A, the powerset of an element or a part of an element of  tar at  A. CLASSES1 th. 13 (Contributed by FL, 13-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
Assertion
Ref Expression
vtarsuelt  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( C  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A )  <->  ( ( C  C_  ( ( tar `  <. X ,  Y >. ) `  A )  /\  C  e.  (
tarskiMap `
 X ) )  \/  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z ) ) ) )
Distinct variable groups:    z, A    z, B    z, C    z, X    z, Y

Proof of Theorem vtarsuelt
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 elin 3358 . . 3  |-  ( C  e.  ( ( { u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u 
C_  z  \/  u  =  ~P z ) }  u.  ~P ( ( tar `  <. X ,  Y >. ) `  A
) )  i^i  ( tarskiMap `  X ) )  <->  ( C  e.  ( { u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u 
C_  z  \/  u  =  ~P z ) }  u.  ~P ( ( tar `  <. X ,  Y >. ) `  A
) )  /\  C  e.  ( tarskiMap `  X )
) )
2 vex 2791 . . . . . . . . . . 11  |-  z  e. 
_V
32ssex 4158 . . . . . . . . . 10  |-  ( C 
C_  z  ->  C  e.  _V )
4 id 19 . . . . . . . . . . 11  |-  ( C  =  ~P z  ->  C  =  ~P z
)
52pwex 4193 . . . . . . . . . . 11  |-  ~P z  e.  _V
64, 5syl6eqel 2371 . . . . . . . . . 10  |-  ( C  =  ~P z  ->  C  e.  _V )
73, 6jaoi 368 . . . . . . . . 9  |-  ( ( C  C_  z  \/  C  =  ~P z
)  ->  C  e.  _V )
87rexlimivw 2663 . . . . . . . 8  |-  ( E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A
) ( C  C_  z  \/  C  =  ~P z )  ->  C  e.  _V )
9 sseq1 3199 . . . . . . . . . 10  |-  ( u  =  C  ->  (
u  C_  z  <->  C  C_  z
) )
10 eqeq1 2289 . . . . . . . . . 10  |-  ( u  =  C  ->  (
u  =  ~P z  <->  C  =  ~P z ) )
119, 10orbi12d 690 . . . . . . . . 9  |-  ( u  =  C  ->  (
( u  C_  z  \/  u  =  ~P z )  <->  ( C  C_  z  \/  C  =  ~P z ) ) )
1211rexbidv 2564 . . . . . . . 8  |-  ( u  =  C  ->  ( E. z  e.  (
( tar `  <. X ,  Y >. ) `  A ) ( u 
C_  z  \/  u  =  ~P z )  <->  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z ) ) )
138, 12elab3 2921 . . . . . . 7  |-  ( C  e.  { u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u 
C_  z  \/  u  =  ~P z ) }  <->  E. z  e.  (
( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z ) )
1413a1i 10 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( C  e.  {
u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u 
C_  z  \/  u  =  ~P z ) }  <->  E. z  e.  (
( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z ) ) )
15 fvex 5539 . . . . . . . 8  |-  ( ( tar `  <. X ,  Y >. ) `  A
)  e.  _V
1615elpw2 4175 . . . . . . 7  |-  ( C  e.  ~P ( ( tar `  <. X ,  Y >. ) `  A
)  <->  C  C_  ( ( tar `  <. X ,  Y >. ) `  A
) )
1716a1i 10 . . . . . 6  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( C  e.  ~P ( ( tar `  <. X ,  Y >. ) `  A )  <->  C  C_  (
( tar `  <. X ,  Y >. ) `  A ) ) )
1814, 17orbi12d 690 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( ( C  e. 
{ u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u  C_  z  \/  u  =  ~P z ) }  \/  C  e.  ~P (
( tar `  <. X ,  Y >. ) `  A ) )  <->  ( E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C  C_  z  \/  C  =  ~P z )  \/  C  C_  ( ( tar `  <. X ,  Y >. ) `  A ) ) ) )
19 elun 3316 . . . . 5  |-  ( C  e.  ( { u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u 
C_  z  \/  u  =  ~P z ) }  u.  ~P ( ( tar `  <. X ,  Y >. ) `  A
) )  <->  ( C  e.  { u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u  C_  z  \/  u  =  ~P z ) }  \/  C  e.  ~P (
( tar `  <. X ,  Y >. ) `  A ) ) )
20 orcom 376 . . . . 5  |-  ( ( C  C_  ( ( tar `  <. X ,  Y >. ) `  A )  \/  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z ) )  <-> 
( E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z )  \/  C  C_  ( ( tar `  <. X ,  Y >. ) `  A ) ) )
2118, 19, 203bitr4g 279 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( C  e.  ( { u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u  C_  z  \/  u  =  ~P z ) }  u.  ~P ( ( tar `  <. X ,  Y >. ) `  A ) )  <->  ( C  C_  ( ( tar `  <. X ,  Y >. ) `  A )  \/  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C  C_  z  \/  C  =  ~P z ) ) ) )
2221anbi1d 685 . . 3  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( ( C  e.  ( { u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u 
C_  z  \/  u  =  ~P z ) }  u.  ~P ( ( tar `  <. X ,  Y >. ) `  A
) )  /\  C  e.  ( tarskiMap `  X )
)  <->  ( ( C 
C_  ( ( tar `  <. X ,  Y >. ) `  A )  \/  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z ) )  /\  C  e.  (
tarskiMap `
 X ) ) ) )
231, 22syl5bb 248 . 2  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( C  e.  ( ( { u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u 
C_  z  \/  u  =  ~P z ) }  u.  ~P ( ( tar `  <. X ,  Y >. ) `  A
) )  i^i  ( tarskiMap `  X ) )  <->  ( ( C  C_  ( ( tar `  <. X ,  Y >. ) `  A )  \/  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z ) )  /\  C  e.  (
tarskiMap `
 X ) ) ) )
24 vtarsu 25298 . . 3  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( ( tar `  <. X ,  Y >. ) `  suc  A )  =  ( ( { u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u 
C_  z  \/  u  =  ~P z ) }  u.  ~P ( ( tar `  <. X ,  Y >. ) `  A
) )  i^i  ( tarskiMap `  X ) ) )
2524eleq2d 2350 . 2  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( C  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A )  <->  C  e.  ( ( { u  |  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( u 
C_  z  \/  u  =  ~P z ) }  u.  ~P ( ( tar `  <. X ,  Y >. ) `  A
) )  i^i  ( tarskiMap `  X ) ) ) )
26 tartarmap 25300 . . . . . . . 8  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( ( tar `  <. X ,  Y >. ) `  A )  C_  ( tarskiMap `  X ) )
2726sselda 3180 . . . . . . 7  |-  ( ( ( X  e.  B  /\  Y  e.  On  /\ 
suc  A  e.  Y
)  /\  z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) )  -> 
z  e.  ( tarskiMap `  X ) )
28 tskmcl 8463 . . . . . . . . . 10  |-  ( tarskiMap `  X )  e.  Tarski
29 tskss 8380 . . . . . . . . . 10  |-  ( ( ( tarskiMap `  X )  e.  Tarski  /\  z  e.  (
tarskiMap `
 X )  /\  C  C_  z )  ->  C  e.  ( tarskiMap `  X
) )
3028, 29mp3an1 1264 . . . . . . . . 9  |-  ( ( z  e.  ( tarskiMap `  X )  /\  C  C_  z )  ->  C  e.  ( tarskiMap `  X )
)
3130ex 423 . . . . . . . 8  |-  ( z  e.  ( tarskiMap `  X
)  ->  ( C  C_  z  ->  C  e.  (
tarskiMap `
 X ) ) )
32 pwtsm 25301 . . . . . . . . 9  |-  ( z  e.  ( tarskiMap `  X
)  ->  ~P z  e.  ( tarskiMap `  X )
)
33 eleq1 2343 . . . . . . . . 9  |-  ( C  =  ~P z  -> 
( C  e.  (
tarskiMap `
 X )  <->  ~P z  e.  ( tarskiMap `  X )
) )
3432, 33syl5ibrcom 213 . . . . . . . 8  |-  ( z  e.  ( tarskiMap `  X
)  ->  ( C  =  ~P z  ->  C  e.  ( tarskiMap `  X )
) )
3531, 34jaod 369 . . . . . . 7  |-  ( z  e.  ( tarskiMap `  X
)  ->  ( ( C  C_  z  \/  C  =  ~P z )  ->  C  e.  ( tarskiMap `  X
) ) )
3627, 35syl 15 . . . . . 6  |-  ( ( ( X  e.  B  /\  Y  e.  On  /\ 
suc  A  e.  Y
)  /\  z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) )  -> 
( ( C  C_  z  \/  C  =  ~P z )  ->  C  e.  ( tarskiMap `  X )
) )
3736rexlimdva 2667 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z )  ->  C  e.  ( tarskiMap `  X
) ) )
3837pm4.71d 615 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z )  <->  ( E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C  C_  z  \/  C  =  ~P z )  /\  C  e.  ( tarskiMap `  X )
) ) )
3938orbi2d 682 . . 3  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( ( ( C 
C_  ( ( tar `  <. X ,  Y >. ) `  A )  /\  C  e.  (
tarskiMap `
 X ) )  \/  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z ) )  <-> 
( ( C  C_  ( ( tar `  <. X ,  Y >. ) `  A )  /\  C  e.  ( tarskiMap `  X )
)  \/  ( E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A
) ( C  C_  z  \/  C  =  ~P z )  /\  C  e.  ( tarskiMap `  X )
) ) ) )
40 andir 838 . . 3  |-  ( ( ( C  C_  (
( tar `  <. X ,  Y >. ) `  A )  \/  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C  C_  z  \/  C  =  ~P z ) )  /\  C  e.  ( tarskiMap `  X
) )  <->  ( ( C  C_  ( ( tar `  <. X ,  Y >. ) `  A )  /\  C  e.  (
tarskiMap `
 X ) )  \/  ( E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z )  /\  C  e.  ( tarskiMap `  X
) ) ) )
4139, 40syl6bbr 254 . 2  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( ( ( C 
C_  ( ( tar `  <. X ,  Y >. ) `  A )  /\  C  e.  (
tarskiMap `
 X ) )  \/  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z ) )  <-> 
( ( C  C_  ( ( tar `  <. X ,  Y >. ) `  A )  \/  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C  C_  z  \/  C  =  ~P z ) )  /\  C  e.  ( tarskiMap `  X
) ) ) )
4223, 25, 413bitr4d 276 1  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( C  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A )  <->  ( ( C  C_  ( ( tar `  <. X ,  Y >. ) `  A )  /\  C  e.  (
tarskiMap `
 X ) )  \/  E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) ( C 
C_  z  \/  C  =  ~P z ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   <.cop 3643   Oncon0 4392   suc csuc 4394   ` cfv 5255   Tarskictsk 8370   tarskiMapctskm 8459   tarctar 25293
This theorem is referenced by:  partarelt1  25308  partarelt2  25309
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-groth 8445
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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-tsk 8371  df-tskm 8460  df-tar 25294
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