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Theorem tskmid 8715
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 20 . . . 4  |-  ( A  e.  x  ->  A  e.  x )
21rgenw 2773 . . 3  |-  A. x  e.  Tarski  ( A  e.  x  ->  A  e.  x )
3 elintrabg 4063 . . 3  |-  ( A  e.  V  ->  ( A  e.  |^| { x  e.  Tarski  |  A  e.  x }  <->  A. x  e.  Tarski  ( A  e.  x  ->  A  e.  x ) ) )
42, 3mpbiri 225 . 2  |-  ( A  e.  V  ->  A  e.  |^| { x  e. 
Tarski  |  A  e.  x } )
5 tskmval 8714 . 2  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
64, 5eleqtrrd 2513 1  |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   A.wral 2705   {crab 2709   |^|cint 4050   ` cfv 5454   Tarskictsk 8623   tarskiMapctskm 8712
This theorem is referenced by:  eltskm  8718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-groth 8698
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-tsk 8624  df-tskm 8713
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