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Theorem tskmval 8461
Description: Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
tskmval  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem tskmval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 grothtsk 8457 . . . . 5  |-  U. Tarski  =  _V
31, 2syl6eleqr 2374 . . . 4  |-  ( A  e.  V  ->  A  e.  U. Tarski )
4 eluni2 3831 . . . 4  |-  ( A  e.  U. Tarski  <->  E. x  e.  Tarski  A  e.  x
)
53, 4sylib 188 . . 3  |-  ( A  e.  V  ->  E. x  e.  Tarski  A  e.  x
)
6 intexrab 4170 . . 3  |-  ( E. x  e.  Tarski  A  e.  x  <->  |^| { x  e. 
Tarski  |  A  e.  x }  e.  _V )
75, 6sylib 188 . 2  |-  ( A  e.  V  ->  |^| { x  e.  Tarski  |  A  e.  x }  e.  _V )
8 eleq1 2343 . . . . 5  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
98rabbidv 2780 . . . 4  |-  ( y  =  A  ->  { x  e.  Tarski  |  y  e.  x }  =  {
x  e.  Tarski  |  A  e.  x } )
109inteqd 3867 . . 3  |-  ( y  =  A  ->  |^| { x  e.  Tarski  |  y  e.  x }  =  |^| { x  e.  Tarski  |  A  e.  x } )
11 df-tskm 8460 . . 3  |-  tarskiMap  =  (
y  e.  _V  |->  |^|
{ x  e.  Tarski  |  y  e.  x }
)
1210, 11fvmptg 5600 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  Tarski  |  A  e.  x }  e.  _V )  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
131, 7, 12syl2anc 642 1  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788   U.cuni 3827   |^|cint 3862   ` cfv 5255   Tarskictsk 8370   tarskiMapctskm 8459
This theorem is referenced by:  tskmid  8462  tskmcl  8463  sstskm  8464  eltskm  8465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-groth 8445
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-tsk 8371  df-tskm 8460
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