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Theorem tskmcl 8721
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 8719 . . 3  |-  ( A  e.  _V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
2 ssrab2 3430 . . . 4  |-  { x  e.  Tarski  |  A  e.  x }  C_  Tarski
3 id 21 . . . . . . 7  |-  ( A  e.  _V  ->  A  e.  _V )
4 grothtsk 8715 . . . . . . 7  |-  U. Tarski  =  _V
53, 4syl6eleqr 2529 . . . . . 6  |-  ( A  e.  _V  ->  A  e.  U. Tarski )
6 eluni2 4021 . . . . . 6  |-  ( A  e.  U. Tarski  <->  E. x  e.  Tarski  A  e.  x
)
75, 6sylib 190 . . . . 5  |-  ( A  e.  _V  ->  E. x  e.  Tarski  A  e.  x
)
8 rabn0 3649 . . . . 5  |-  ( { x  e.  Tarski  |  A  e.  x }  =/=  (/)  <->  E. x  e.  Tarski  A  e.  x
)
97, 8sylibr 205 . . . 4  |-  ( A  e.  _V  ->  { x  e.  Tarski  |  A  e.  x }  =/=  (/) )
10 inttsk 8654 . . . 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 646 . . 3  |-  ( A  e.  _V  ->  |^| { x  e.  Tarski  |  A  e.  x }  e.  Tarski )
121, 11eqeltrd 2512 . 2  |-  ( A  e.  _V  ->  ( tarskiMap `  A )  e.  Tarski )
13 fvprc 5725 . . 3  |-  ( -.  A  e.  _V  ->  (
tarskiMap `
 A )  =  (/) )
14 0tsk 8635 . . 3  |-  (/)  e.  Tarski
1513, 14syl6eqel 2526 . 2  |-  ( -.  A  e.  _V  ->  (
tarskiMap `
 A )  e. 
Tarski )
1612, 15pm2.61i 159 1  |-  ( tarskiMap `  A )  e.  Tarski
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1726    =/= wne 2601   E.wrex 2708   {crab 2711   _Vcvv 2958    C_ wss 3322   (/)c0 3630   U.cuni 4017   |^|cint 4052   ` cfv 5457   Tarskictsk 8628   tarskiMapctskm 8717
This theorem is referenced by:  eltskm  8723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-groth 8703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-er 6908  df-en 7113  df-dom 7114  df-tsk 8629  df-tskm 8718
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