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Theorem tskmval 8714
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 2964 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 grothtsk 8710 . . . . 5  |-  U. Tarski  =  _V
31, 2syl6eleqr 2527 . . . 4  |-  ( A  e.  V  ->  A  e.  U. Tarski )
4 eluni2 4019 . . . 4  |-  ( A  e.  U. Tarski  <->  E. x  e.  Tarski  A  e.  x
)
53, 4sylib 189 . . 3  |-  ( A  e.  V  ->  E. x  e.  Tarski  A  e.  x
)
6 intexrab 4359 . . 3  |-  ( E. x  e.  Tarski  A  e.  x  <->  |^| { x  e. 
Tarski  |  A  e.  x }  e.  _V )
75, 6sylib 189 . 2  |-  ( A  e.  V  ->  |^| { x  e.  Tarski  |  A  e.  x }  e.  _V )
8 eleq1 2496 . . . . 5  |-  ( y  =  A  ->  (
y  e.  x  <->  A  e.  x ) )
98rabbidv 2948 . . . 4  |-  ( y  =  A  ->  { x  e.  Tarski  |  y  e.  x }  =  {
x  e.  Tarski  |  A  e.  x } )
109inteqd 4055 . . 3  |-  ( y  =  A  ->  |^| { x  e.  Tarski  |  y  e.  x }  =  |^| { x  e.  Tarski  |  A  e.  x } )
11 df-tskm 8713 . . 3  |-  tarskiMap  =  (
y  e.  _V  |->  |^|
{ x  e.  Tarski  |  y  e.  x }
)
1210, 11fvmptg 5804 . 2  |-  ( ( A  e.  _V  /\  |^|
{ x  e.  Tarski  |  A  e.  x }  e.  _V )  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
131, 7, 12syl2anc 643 1  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   E.wrex 2706   {crab 2709   _Vcvv 2956   U.cuni 4015   |^|cint 4050   ` cfv 5454   Tarskictsk 8623   tarskiMapctskm 8712
This theorem is referenced by:  tskmid  8715  tskmcl  8716  sstskm  8717  eltskm  8718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-groth 8698
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-tsk 8624  df-tskm 8713
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