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Theorem tskmid 9105
Description: The set  A is an element of the smallest Tarski 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 22 . . . 4  |-  ( A  e.  x  ->  A  e.  x )
21rgenw 2888 . . 3  |-  A. x  e.  Tarski  ( A  e.  x  ->  A  e.  x )
3 elintrabg 4236 . . 3  |-  ( A  e.  V  ->  ( A  e.  |^| { x  e.  Tarski  |  A  e.  x }  <->  A. x  e.  Tarski  ( A  e.  x  ->  A  e.  x ) ) )
42, 3mpbiri 233 . 2  |-  ( A  e.  V  ->  A  e.  |^| { x  e. 
Tarski  |  A  e.  x } )
5 tskmval 9104 . 2  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
64, 5eleqtrrd 2540 1  |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   A.wral 2793   {crab 2797   |^|cint 4223   ` cfv 5513   Tarskictsk 9013   tarskiMapctskm 9102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-groth 9088
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-int 4224  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-iota 5476  df-fun 5515  df-fv 5521  df-tsk 9014  df-tskm 9103
This theorem is referenced by:  eltskm  9108
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