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Theorem tskmval 9206
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 3115 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 grothtsk 9202 . . . . 5  |-  U. Tarski  =  _V
31, 2syl6eleqr 2553 . . . 4  |-  ( A  e.  V  ->  A  e.  U. Tarski )
4 eluni2 4239 . . . 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 4596 . . 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 2526 . . . . 5  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
98rabbidv 3098 . . . 4  |-  ( y  =  A  ->  { x  e.  Tarski  |  y  e.  x }  =  {
x  e.  Tarski  |  A  e.  x } )
109inteqd 4276 . . 3  |-  ( y  =  A  ->  |^| { x  e.  Tarski  |  y  e.  x }  =  |^| { x  e.  Tarski  |  A  e.  x } )
11 df-tskm 9205 . . 3  |-  tarskiMap  =  (
y  e.  _V  |->  |^|
{ x  e.  Tarski  |  y  e.  x }
)
1210, 11fvmptg 5929 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  Tarski  |  A  e.  x }  e.  _V )  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
131, 7, 12syl2anc 659 1  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   E.wrex 2805   {crab 2808   _Vcvv 3106   U.cuni 4235   |^|cint 4271   ` cfv 5570   Tarskictsk 9115   tarskiMapctskm 9204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-groth 9190
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-tsk 9116  df-tskm 9205
This theorem is referenced by:  tskmid  9207  tskmcl  9208  sstskm  9209  eltskm  9210
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