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Theorem tskmcl 8463
Description: A Tarski's class that contains  A is a Tarski's class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
Assertion
Ref Expression
tskmcl  |-  ( tarskiMap `  A )  e.  Tarski

Proof of Theorem tskmcl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 tskmval 8461 . . 3  |-  ( A  e.  _V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
2 ssrab2 3258 . . . 4  |-  { x  e.  Tarski  |  A  e.  x }  C_  Tarski
3 id 19 . . . . . . 7  |-  ( A  e.  _V  ->  A  e.  _V )
4 grothtsk 8457 . . . . . . 7  |-  U. Tarski  =  _V
53, 4syl6eleqr 2374 . . . . . 6  |-  ( A  e.  _V  ->  A  e.  U. Tarski )
6 eluni2 3831 . . . . . 6  |-  ( A  e.  U. Tarski  <->  E. x  e.  Tarski  A  e.  x
)
75, 6sylib 188 . . . . 5  |-  ( A  e.  _V  ->  E. x  e.  Tarski  A  e.  x
)
8 rabn0 3474 . . . . 5  |-  ( { x  e.  Tarski  |  A  e.  x }  =/=  (/)  <->  E. x  e.  Tarski  A  e.  x
)
97, 8sylibr 203 . . . 4  |-  ( A  e.  _V  ->  { x  e.  Tarski  |  A  e.  x }  =/=  (/) )
10 inttsk 8396 . . . 4  |-  ( ( { x  e.  Tarski  |  A  e.  x }  C_ 
Tarski  /\  { x  e. 
Tarski  |  A  e.  x }  =/=  (/) )  ->  |^| { x  e.  Tarski  |  A  e.  x }  e.  Tarski )
112, 9, 10sylancr 644 . . 3  |-  ( A  e.  _V  ->  |^| { x  e.  Tarski  |  A  e.  x }  e.  Tarski )
121, 11eqeltrd 2357 . 2  |-  ( A  e.  _V  ->  ( tarskiMap `  A )  e.  Tarski )
13 fvprc 5519 . . 3  |-  ( -.  A  e.  _V  ->  (
tarskiMap `
 A )  =  (/) )
14 0tsk 8377 . . 3  |-  (/)  e.  Tarski
1513, 14syl6eqel 2371 . 2  |-  ( -.  A  e.  _V  ->  (
tarskiMap `
 A )  e. 
Tarski )
1612, 15pm2.61i 156 1  |-  ( tarskiMap `  A )  e.  Tarski
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   (/)c0 3455   U.cuni 3827   |^|cint 3862   ` cfv 5255   Tarskictsk 8370   tarskiMapctskm 8459
This theorem is referenced by:  eltskm  8465  pwtsm  25889  subtsm  25890  subtareqbe  25891  vtarsuelt  25895
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-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-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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-er 6660  df-en 6864  df-dom 6865  df-tsk 8371  df-tskm 8460
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