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Theorem vtarsuelt 25998
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 3371 . . 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 2804 . . . . . . . . . . 11  |-  z  e. 
_V
32ssex 4174 . . . . . . . . . 10  |-  ( C 
C_  z  ->  C  e.  _V )
4 id 19 . . . . . . . . . . 11  |-  ( C  =  ~P z  ->  C  =  ~P z
)
52pwex 4209 . . . . . . . . . . 11  |-  ~P z  e.  _V
64, 5syl6eqel 2384 . . . . . . . . . 10  |-  ( C  =  ~P z  ->  C  e.  _V )
73, 6jaoi 368 . . . . . . . . 9  |-  ( ( C  C_  z  \/  C  =  ~P z
)  ->  C  e.  _V )
87rexlimivw 2676 . . . . . . . 8  |-  ( E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A
) ( C  C_  z  \/  C  =  ~P z )  ->  C  e.  _V )
9 sseq1 3212 . . . . . . . . . 10  |-  ( u  =  C  ->  (
u  C_  z  <->  C  C_  z
) )
10 eqeq1 2302 . . . . . . . . . 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 2577 . . . . . . . 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 2934 . . . . . . 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 5555 . . . . . . . 8  |-  ( ( tar `  <. X ,  Y >. ) `  A
)  e.  _V
1615elpw2 4191 . . . . . . 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 3329 . . . . 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 25989 . . 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 2363 . 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 25991 . . . . . . . 8  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( ( tar `  <. X ,  Y >. ) `  A )  C_  ( tarskiMap `  X ) )
2726sselda 3193 . . . . . . 7  |-  ( ( ( X  e.  B  /\  Y  e.  On  /\ 
suc  A  e.  Y
)  /\  z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) )  -> 
z  e.  ( tarskiMap `  X ) )
28 tskmcl 8479 . . . . . . . . . 10  |-  ( tarskiMap `  X )  e.  Tarski
29 tskss 8396 . . . . . . . . . 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 25992 . . . . . . . . 9  |-  ( z  e.  ( tarskiMap `  X
)  ->  ~P z  e.  ( tarskiMap `  X )
)
33 eleq1 2356 . . . . . . . . 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 2680 . . . . 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 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801    u. cun 3163    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   <.cop 3656   Oncon0 4408   suc csuc 4410   ` cfv 5271   Tarskictsk 8386   tarskiMapctskm 8475   tarctar 25984
This theorem is referenced by:  partarelt1  25999  partarelt2  26000
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-groth 8461
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  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-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-tsk 8387  df-tskm 8476  df-tar 25985
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