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Theorem eltskm 8720
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 8716 . . 3  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
21eleq2d 2505 . 2  |-  ( A  e.  V  ->  ( B  e.  ( tarskiMap `  A
)  <->  B  e.  |^| { x  e.  Tarski  |  A  e.  x } ) )
3 elex 2966 . . . 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 8717 . . . . 5  |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A )
)
6 tskmcl 8718 . . . . . 6  |-  ( tarskiMap `  A )  e.  Tarski
7 eleq2 2499 . . . . . . . 8  |-  ( x  =  ( tarskiMap `  A
)  ->  ( A  e.  x  <->  A  e.  ( tarskiMap `  A ) ) )
8 eleq2 2499 . . . . . . . 8  |-  ( x  =  ( tarskiMap `  A
)  ->  ( B  e.  x  <->  B  e.  ( tarskiMap `  A ) ) )
97, 8imbi12d 313 . . . . . . 7  |-  ( x  =  ( tarskiMap `  A
)  ->  ( ( A  e.  x  ->  B  e.  x )  <->  ( A  e.  ( tarskiMap `  A )  ->  B  e.  ( tarskiMap `  A ) ) ) )
109rspcv 3050 . . . . . 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 8 . . . . 5  |-  ( A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )  ->  ( A  e.  ( tarskiMap `  A
)  ->  B  e.  (
tarskiMap `
 A ) ) )
125, 11syl5com 29 . . . 4  |-  ( A  e.  V  ->  ( A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )  ->  B  e.  ( tarskiMap `  A )
) )
13 elex 2966 . . . 4  |-  ( B  e.  ( tarskiMap `  A
)  ->  B  e.  _V )
1412, 13syl6 32 . . 3  |-  ( A  e.  V  ->  ( A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )  ->  B  e.  _V ) )
15 elintrabg 4065 . . . 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 345 . 2  |-  ( A  e.  V  ->  ( B  e.  |^| { x  e.  Tarski  |  A  e.  x }  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x ) ) )
182, 17bitrd 246 1  |-  ( A  e.  V  ->  ( B  e.  ( tarskiMap `  A
)  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958   |^|cint 4052   ` cfv 5456   Tarskictsk 8625   tarskiMapctskm 8714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-groth 8700
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-er 6907  df-en 7112  df-dom 7113  df-tsk 8626  df-tskm 8715
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