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Theorem tskmval 9109
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 3079 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 grothtsk 9105 . . . . 5  |-  U. Tarski  =  _V
31, 2syl6eleqr 2550 . . . 4  |-  ( A  e.  V  ->  A  e.  U. Tarski )
4 eluni2 4195 . . . 4  |-  ( A  e.  U. Tarski  <->  E. x  e.  Tarski  A  e.  x
)
53, 4sylib 196 . . 3  |-  ( A  e.  V  ->  E. x  e.  Tarski  A  e.  x
)
6 intexrab 4551 . . 3  |-  ( E. x  e.  Tarski  A  e.  x  <->  |^| { x  e. 
Tarski  |  A  e.  x }  e.  _V )
75, 6sylib 196 . 2  |-  ( A  e.  V  ->  |^| { x  e.  Tarski  |  A  e.  x }  e.  _V )
8 eleq1 2523 . . . . 5  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
98rabbidv 3062 . . . 4  |-  ( y  =  A  ->  { x  e.  Tarski  |  y  e.  x }  =  {
x  e.  Tarski  |  A  e.  x } )
109inteqd 4233 . . 3  |-  ( y  =  A  ->  |^| { x  e.  Tarski  |  y  e.  x }  =  |^| { x  e.  Tarski  |  A  e.  x } )
11 df-tskm 9108 . . 3  |-  tarskiMap  =  (
y  e.  _V  |->  |^|
{ x  e.  Tarski  |  y  e.  x }
)
1210, 11fvmptg 5873 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  Tarski  |  A  e.  x }  e.  _V )  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
131, 7, 12syl2anc 661 1  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   E.wrex 2796   {crab 2799   _Vcvv 3070   U.cuni 4191   |^|cint 4228   ` cfv 5518   Tarskictsk 9018   tarskiMapctskm 9107
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 4513  ax-nul 4521  ax-pr 4631  ax-groth 9093
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 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-int 4229  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-tsk 9019  df-tskm 9108
This theorem is referenced by:  tskmid  9110  tskmcl  9111  sstskm  9112  eltskm  9113
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