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Theorem tskmid 8478
Description: The set  A is an element of the smallest Tarski's class that contains  A. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
Assertion
Ref Expression
tskmid  |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A )
)

Proof of Theorem tskmid
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4  |-  ( A  e.  x  ->  A  e.  x )
21rgenw 2623 . . 3  |-  A. x  e.  Tarski  ( A  e.  x  ->  A  e.  x )
3 elintrabg 3891 . . 3  |-  ( A  e.  V  ->  ( A  e.  |^| { x  e.  Tarski  |  A  e.  x }  <->  A. x  e.  Tarski  ( A  e.  x  ->  A  e.  x ) ) )
42, 3mpbiri 224 . 2  |-  ( A  e.  V  ->  A  e.  |^| { x  e. 
Tarski  |  A  e.  x } )
5 tskmval 8477 . 2  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
64, 5eleqtrrd 2373 1  |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   A.wral 2556   {crab 2560   |^|cint 3878   ` cfv 5271   Tarskictsk 8386   tarskiMapctskm 8475
This theorem is referenced by:  eltskm  8481  tartarmap  25991
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-pr 4230  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-iota 5235  df-fun 5273  df-fv 5279  df-tsk 8387  df-tskm 8476
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