MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tskmcl Unicode version

Theorem tskmcl 8479
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 8477 . . 3  |-  ( A  e.  _V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
2 ssrab2 3271 . . . 4  |-  { x  e.  Tarski  |  A  e.  x }  C_  Tarski
3 id 19 . . . . . . 7  |-  ( A  e.  _V  ->  A  e.  _V )
4 grothtsk 8473 . . . . . . 7  |-  U. Tarski  =  _V
53, 4syl6eleqr 2387 . . . . . 6  |-  ( A  e.  _V  ->  A  e.  U. Tarski )
6 eluni2 3847 . . . . . 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 3487 . . . . 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 8412 . . . 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 2370 . 2  |-  ( A  e.  _V  ->  ( tarskiMap `  A )  e.  Tarski )
13 fvprc 5535 . . 3  |-  ( -.  A  e.  _V  ->  (
tarskiMap `
 A )  =  (/) )
14 0tsk 8393 . . 3  |-  (/)  e.  Tarski
1513, 14syl6eqel 2384 . 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 1696    =/= wne 2459   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   U.cuni 3843   |^|cint 3878   ` cfv 5271   Tarskictsk 8386   tarskiMapctskm 8475
This theorem is referenced by:  eltskm  8481  pwtsm  25992  subtsm  25993  subtareqbe  25994  vtarsuelt  25998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-groth 8461
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-tsk 8387  df-tskm 8476
  Copyright terms: Public domain W3C validator