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Theorem tskmcl 9111
Description: A Tarski class that contains  A is a Tarski 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 9109 . . 3  |-  ( A  e.  _V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
2 ssrab2 3537 . . . 4  |-  { x  e.  Tarski  |  A  e.  x }  C_  Tarski
3 id 22 . . . . . . 7  |-  ( A  e.  _V  ->  A  e.  _V )
4 grothtsk 9105 . . . . . . 7  |-  U. Tarski  =  _V
53, 4syl6eleqr 2550 . . . . . 6  |-  ( A  e.  _V  ->  A  e.  U. Tarski )
6 eluni2 4195 . . . . . 6  |-  ( A  e.  U. Tarski  <->  E. x  e.  Tarski  A  e.  x
)
75, 6sylib 196 . . . . 5  |-  ( A  e.  _V  ->  E. x  e.  Tarski  A  e.  x
)
8 rabn0 3757 . . . . 5  |-  ( { x  e.  Tarski  |  A  e.  x }  =/=  (/)  <->  E. x  e.  Tarski  A  e.  x
)
97, 8sylibr 212 . . . 4  |-  ( A  e.  _V  ->  { x  e.  Tarski  |  A  e.  x }  =/=  (/) )
10 inttsk 9044 . . . 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 663 . . 3  |-  ( A  e.  _V  ->  |^| { x  e.  Tarski  |  A  e.  x }  e.  Tarski )
121, 11eqeltrd 2539 . 2  |-  ( A  e.  _V  ->  ( tarskiMap `  A )  e.  Tarski )
13 fvprc 5785 . . 3  |-  ( -.  A  e.  _V  ->  (
tarskiMap `
 A )  =  (/) )
14 0tsk 9025 . . 3  |-  (/)  e.  Tarski
1513, 14syl6eqel 2547 . 2  |-  ( -.  A  e.  _V  ->  (
tarskiMap `
 A )  e. 
Tarski )
1612, 15pm2.61i 164 1  |-  ( tarskiMap `  A )  e.  Tarski
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    e. wcel 1758    =/= wne 2644   E.wrex 2796   {crab 2799   _Vcvv 3070    C_ wss 3428   (/)c0 3737   U.cuni 4191   |^|cint 4228   ` cfv 5518   Tarskictsk 9018   tarskiMapctskm 9107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-groth 9093
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-int 4229  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-er 7203  df-en 7413  df-dom 7414  df-tsk 9019  df-tskm 9108
This theorem is referenced by:  eltskm  9113
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