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Theorem tskmcl 8642
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 8640 . . 3  |-  ( A  e.  _V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
2 ssrab2 3364 . . . 4  |-  { x  e.  Tarski  |  A  e.  x }  C_  Tarski
3 id 20 . . . . . . 7  |-  ( A  e.  _V  ->  A  e.  _V )
4 grothtsk 8636 . . . . . . 7  |-  U. Tarski  =  _V
53, 4syl6eleqr 2471 . . . . . 6  |-  ( A  e.  _V  ->  A  e.  U. Tarski )
6 eluni2 3954 . . . . . 6  |-  ( A  e.  U. Tarski  <->  E. x  e.  Tarski  A  e.  x
)
75, 6sylib 189 . . . . 5  |-  ( A  e.  _V  ->  E. x  e.  Tarski  A  e.  x
)
8 rabn0 3583 . . . . 5  |-  ( { x  e.  Tarski  |  A  e.  x }  =/=  (/)  <->  E. x  e.  Tarski  A  e.  x
)
97, 8sylibr 204 . . . 4  |-  ( A  e.  _V  ->  { x  e.  Tarski  |  A  e.  x }  =/=  (/) )
10 inttsk 8575 . . . 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 645 . . 3  |-  ( A  e.  _V  ->  |^| { x  e.  Tarski  |  A  e.  x }  e.  Tarski )
121, 11eqeltrd 2454 . 2  |-  ( A  e.  _V  ->  ( tarskiMap `  A )  e.  Tarski )
13 fvprc 5655 . . 3  |-  ( -.  A  e.  _V  ->  (
tarskiMap `
 A )  =  (/) )
14 0tsk 8556 . . 3  |-  (/)  e.  Tarski
1513, 14syl6eqel 2468 . 2  |-  ( -.  A  e.  _V  ->  (
tarskiMap `
 A )  e. 
Tarski )
1612, 15pm2.61i 158 1  |-  ( tarskiMap `  A )  e.  Tarski
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1717    =/= wne 2543   E.wrex 2643   {crab 2646   _Vcvv 2892    C_ wss 3256   (/)c0 3564   U.cuni 3950   |^|cint 3985   ` cfv 5387   Tarskictsk 8549   tarskiMapctskm 8638
This theorem is referenced by:  eltskm  8644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-groth 8624
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-int 3986  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-er 6834  df-en 7039  df-dom 7040  df-tsk 8550  df-tskm 8639
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