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Theorem tskmval 8477
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 2809 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 grothtsk 8473 . . . . 5  |-  U. Tarski  =  _V
31, 2syl6eleqr 2387 . . . 4  |-  ( A  e.  V  ->  A  e.  U. Tarski )
4 eluni2 3847 . . . 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 4186 . . 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 2356 . . . . 5  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
98rabbidv 2793 . . . 4  |-  ( y  =  A  ->  { x  e.  Tarski  |  y  e.  x }  =  {
x  e.  Tarski  |  A  e.  x } )
109inteqd 3883 . . 3  |-  ( y  =  A  ->  |^| { x  e.  Tarski  |  y  e.  x }  =  |^| { x  e.  Tarski  |  A  e.  x } )
11 df-tskm 8476 . . 3  |-  tarskiMap  =  (
y  e.  _V  |->  |^|
{ x  e.  Tarski  |  y  e.  x }
)
1210, 11fvmptg 5616 . 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 1632    e. wcel 1696   E.wrex 2557   {crab 2560   _Vcvv 2801   U.cuni 3843   |^|cint 3878   ` cfv 5271   Tarskictsk 8386   tarskiMapctskm 8475
This theorem is referenced by:  tskmid  8478  tskmcl  8479  sstskm  8480  eltskm  8481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-groth 8461
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-tsk 8387  df-tskm 8476
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