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Theorem eltskm 8510
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 8506 . . 3  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
21eleq2d 2383 . 2  |-  ( A  e.  V  ->  ( B  e.  ( tarskiMap `  A
)  <->  B  e.  |^| { x  e.  Tarski  |  A  e.  x } ) )
3 elex 2830 . . . 4  |-  ( B  e.  |^| { x  e. 
Tarski  |  A  e.  x }  ->  B  e.  _V )
43a1i 10 . . 3  |-  ( A  e.  V  ->  ( B  e.  |^| { x  e.  Tarski  |  A  e.  x }  ->  B  e.  _V ) )
5 tskmid 8507 . . . . 5  |-  ( A  e.  V  ->  A  e.  ( tarskiMap `  A )
)
6 tskmcl 8508 . . . . . 6  |-  ( tarskiMap `  A )  e.  Tarski
7 eleq2 2377 . . . . . . . 8  |-  ( x  =  ( tarskiMap `  A
)  ->  ( A  e.  x  <->  A  e.  ( tarskiMap `  A ) ) )
8 eleq2 2377 . . . . . . . 8  |-  ( x  =  ( tarskiMap `  A
)  ->  ( B  e.  x  <->  B  e.  ( tarskiMap `  A ) ) )
97, 8imbi12d 311 . . . . . . 7  |-  ( x  =  ( tarskiMap `  A
)  ->  ( ( A  e.  x  ->  B  e.  x )  <->  ( A  e.  ( tarskiMap `  A )  ->  B  e.  ( tarskiMap `  A ) ) ) )
109rspcv 2914 . . . . . 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 26 . . . 4  |-  ( A  e.  V  ->  ( A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )  ->  B  e.  ( tarskiMap `  A )
) )
13 elex 2830 . . . 4  |-  ( B  e.  ( tarskiMap `  A
)  ->  B  e.  _V )
1412, 13syl6 29 . . 3  |-  ( A  e.  V  ->  ( A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x )  ->  B  e.  _V ) )
15 elintrabg 3912 . . . 4  |-  ( B  e.  _V  ->  ( B  e.  |^| { x  e.  Tarski  |  A  e.  x }  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x ) ) )
1615a1i 10 . . 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 343 . 2  |-  ( A  e.  V  ->  ( B  e.  |^| { x  e.  Tarski  |  A  e.  x }  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  e.  x ) ) )
182, 17bitrd 244 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 176    = wceq 1633    e. wcel 1701   A.wral 2577   {crab 2581   _Vcvv 2822   |^|cint 3899   ` cfv 5292   Tarskictsk 8415   tarskiMapctskm 8504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-groth 8490
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-er 6702  df-en 6907  df-dom 6908  df-tsk 8416  df-tskm 8505
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