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Theorem partarelt1 25896
Description: If  C is a part of an element of our tar function at  A then  C is an element or tar at 
suc  A. CLASSES1 th. 14 (Contributed by FL, 13-Apr-2011.)
Assertion
Ref Expression
partarelt1  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( ( C  C_  Z  /\  Z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) )  ->  C  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A
) ) )

Proof of Theorem partarelt1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 orc 374 . . 3  |-  ( C 
C_  Z  ->  ( C  C_  Z  \/  C  =  ~P Z ) )
2 sseq2 3200 . . . . . . 7  |-  ( z  =  Z  ->  ( C  C_  z  <->  C  C_  Z
) )
3 pweq 3628 . . . . . . . 8  |-  ( z  =  Z  ->  ~P z  =  ~P Z
)
43eqeq2d 2294 . . . . . . 7  |-  ( z  =  Z  ->  ( C  =  ~P z  <->  C  =  ~P Z ) )
52, 4orbi12d 690 . . . . . 6  |-  ( z  =  Z  ->  (
( C  C_  z  \/  C  =  ~P z )  <->  ( C  C_  Z  \/  C  =  ~P Z ) ) )
65rspcev 2884 . . . . 5  |-  ( ( 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 ) )
7 olc 373 . . . . . 6  |-  ( 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 ) ) )
8 vtarsuelt 25895 . . . . . 6  |-  ( ( 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 ) ) ) )
97, 8syl5ibrcom 213 . . . . 5  |-  ( E. z  e.  ( ( tar `  <. X ,  Y >. ) `  A
) ( C  C_  z  \/  C  =  ~P z )  ->  (
( X  e.  B  /\  Y  e.  On  /\ 
suc  A  e.  Y
)  ->  C  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A ) ) )
106, 9syl 15 . . . 4  |-  ( ( Z  e.  ( ( tar `  <. X ,  Y >. ) `  A
)  /\  ( C  C_  Z  \/  C  =  ~P Z ) )  ->  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  ->  C  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A ) ) )
1110ex 423 . . 3  |-  ( Z  e.  ( ( tar `  <. X ,  Y >. ) `  A )  ->  ( ( C 
C_  Z  \/  C  =  ~P Z )  -> 
( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  ->  C  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A ) ) ) )
121, 11mpan9 455 . 2  |-  ( ( C  C_  Z  /\  Z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) )  ->  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  ->  C  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A
) ) )
1312com12 27 1  |-  ( ( X  e.  B  /\  Y  e.  On  /\  suc  A  e.  Y )  -> 
( ( C  C_  Z  /\  Z  e.  ( ( tar `  <. X ,  Y >. ) `  A ) )  ->  C  e.  ( ( tar `  <. X ,  Y >. ) `  suc  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ~Pcpw 3625   <.cop 3643   Oncon0 4392   suc csuc 4394   ` cfv 5255   tarskiMapctskm 8459   tarctar 25881
This theorem is referenced by:  tareltsuc  25898
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 25882
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