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Theorem sstskm 8480
Description: Being a part of  ( tarskiMap `  A ). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
Assertion
Ref Expression
sstskm  |-  ( A  e.  V  ->  ( B  C_  ( tarskiMap `  A
)  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  C_  x ) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem sstskm
StepHypRef Expression
1 tskmval 8477 . . . 4  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
2 df-rab 2565 . . . . 5  |-  { x  e.  Tarski  |  A  e.  x }  =  {
x  |  ( x  e.  Tarski  /\  A  e.  x ) }
32inteqi 3882 . . . 4  |-  |^| { x  e.  Tarski  |  A  e.  x }  =  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) }
41, 3syl6eq 2344 . . 3  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) } )
54sseq2d 3219 . 2  |-  ( A  e.  V  ->  ( B  C_  ( tarskiMap `  A
)  <->  B  C_  |^| { x  |  ( x  e. 
Tarski  /\  A  e.  x
) } ) )
6 impexp 433 . . . 4  |-  ( ( ( x  e.  Tarski  /\  A  e.  x )  ->  B  C_  x
)  <->  ( x  e. 
Tarski  ->  ( A  e.  x  ->  B  C_  x
) ) )
76albii 1556 . . 3  |-  ( A. x ( ( x  e.  Tarski  /\  A  e.  x )  ->  B  C_  x )  <->  A. x
( x  e.  Tarski  -> 
( A  e.  x  ->  B  C_  x )
) )
8 ssintab 3895 . . 3  |-  ( B 
C_  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) }  <->  A. x ( ( x  e.  Tarski  /\  A  e.  x )  ->  B  C_  x ) )
9 df-ral 2561 . . 3  |-  ( A. x  e.  Tarski  ( A  e.  x  ->  B  C_  x )  <->  A. x
( x  e.  Tarski  -> 
( A  e.  x  ->  B  C_  x )
) )
107, 8, 93bitr4i 268 . 2  |-  ( B 
C_  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) }  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  C_  x ) )
115, 10syl6bb 252 1  |-  ( A  e.  V  ->  ( B  C_  ( tarskiMap `  A
)  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  C_  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    e. wcel 1696   {cab 2282   A.wral 2556   {crab 2560    C_ wss 3165   |^|cint 3878   ` cfv 5271   Tarskictsk 8386   tarskiMapctskm 8475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-groth 8461
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-tsk 8387  df-tskm 8476
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