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Theorem eltskm 9233
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 9229 . . 3  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
21eleq2d 2537 . 2  |-  ( A  e.  V  ->  ( B  e.  ( tarskiMap `  A
)  <->  B  e.  |^| { x  e.  Tarski  |  A  e.  x } ) )
3 elex 3127 . . . 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 9230 . . . . 5  |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A )
)
6 tskmcl 9231 . . . . . 6  |-  ( tarskiMap `  A )  e.  Tarski
7 eleq2 2540 . . . . . . . 8  |-  ( x  =  ( tarskiMap `  A
)  ->  ( A  e.  x  <->  A  e.  ( tarskiMap `  A ) ) )
8 eleq2 2540 . . . . . . . 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 3215 . . . . . 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 3127 . . . 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 4301 . . . 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 1379    e. wcel 1767   A.wral 2817   {crab 2821   _Vcvv 3118   |^|cint 4288   ` cfv 5594   Tarskictsk 9138   tarskiMapctskm 9227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-groth 9213
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7323  df-en 7529  df-dom 7530  df-tsk 9139  df-tskm 9228
This theorem is referenced by: (None)
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