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Theorem eltskm 9010
Description: Belonging to  ( tarskiMap `  A ). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
Assertion
Ref Expression
eltskm  |-  ( A  e.  V  ->  ( B  e.  ( tarskiMap `  A
)  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )
) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem eltskm
StepHypRef Expression
1 tskmval 9006 . . 3  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
21eleq2d 2510 . 2  |-  ( A  e.  V  ->  ( B  e.  ( tarskiMap `  A
)  <->  B  e.  |^| { x  e.  Tarski  |  A  e.  x } ) )
3 elex 2981 . . . 4  |-  ( B  e.  |^| { x  e. 
Tarski  |  A  e.  x }  ->  B  e.  _V )
43a1i 11 . . 3  |-  ( A  e.  V  ->  ( B  e.  |^| { x  e.  Tarski  |  A  e.  x }  ->  B  e.  _V ) )
5 tskmid 9007 . . . . 5  |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A )
)
6 tskmcl 9008 . . . . . 6  |-  ( tarskiMap `  A )  e.  Tarski
7 eleq2 2504 . . . . . . . 8  |-  ( x  =  ( tarskiMap `  A
)  ->  ( A  e.  x  <->  A  e.  ( tarskiMap `  A ) ) )
8 eleq2 2504 . . . . . . . 8  |-  ( x  =  ( tarskiMap `  A
)  ->  ( B  e.  x  <->  B  e.  ( tarskiMap `  A ) ) )
97, 8imbi12d 320 . . . . . . 7  |-  ( x  =  ( tarskiMap `  A
)  ->  ( ( A  e.  x  ->  B  e.  x )  <->  ( A  e.  ( tarskiMap `  A )  ->  B  e.  ( tarskiMap `  A ) ) ) )
109rspcv 3069 . . . . . 6  |-  ( (
tarskiMap `
 A )  e. 
Tarski  ->  ( A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )  ->  ( A  e.  ( tarskiMap `  A
)  ->  B  e.  (
tarskiMap `
 A ) ) ) )
116, 10ax-mp 5 . . . . 5  |-  ( A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )  ->  ( A  e.  ( tarskiMap `  A
)  ->  B  e.  (
tarskiMap `
 A ) ) )
125, 11syl5com 30 . . . 4  |-  ( A  e.  V  ->  ( A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )  ->  B  e.  ( tarskiMap `  A )
) )
13 elex 2981 . . . 4  |-  ( B  e.  ( tarskiMap `  A
)  ->  B  e.  _V )
1412, 13syl6 33 . . 3  |-  ( A  e.  V  ->  ( A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )  ->  B  e.  _V ) )
15 elintrabg 4141 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  |^| { x  e.  Tarski  |  A  e.  x }  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x ) ) )
1615a1i 11 . . 3  |-  ( A  e.  V  ->  ( B  e.  _V  ->  ( B  e.  |^| { x  e.  Tarski  |  A  e.  x }  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x ) ) ) )
174, 14, 16pm5.21ndd 354 . 2  |-  ( A  e.  V  ->  ( B  e.  |^| { x  e.  Tarski  |  A  e.  x }  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x ) ) )
182, 17bitrd 253 1  |-  ( A  e.  V  ->  ( B  e.  ( tarskiMap `  A
)  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719   _Vcvv 2972   |^|cint 4128   ` cfv 5418   Tarskictsk 8915   tarskiMapctskm 9004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-groth 8990
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-er 7101  df-en 7311  df-dom 7312  df-tsk 8916  df-tskm 9005
This theorem is referenced by: (None)
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