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Theorem sstskm 8717
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 8714 . . . 4  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
2 df-rab 2714 . . . . 5  |-  { x  e.  Tarski  |  A  e.  x }  =  {
x  |  ( x  e.  Tarski  /\  A  e.  x ) }
32inteqi 4054 . . . 4  |-  |^| { x  e.  Tarski  |  A  e.  x }  =  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) }
41, 3syl6eq 2484 . . 3  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) } )
54sseq2d 3376 . 2  |-  ( A  e.  V  ->  ( B  C_  ( tarskiMap `  A
)  <->  B  C_  |^| { x  |  ( x  e. 
Tarski  /\  A  e.  x
) } ) )
6 impexp 434 . . . 4  |-  ( ( ( x  e.  Tarski  /\  A  e.  x )  ->  B  C_  x
)  <->  ( x  e. 
Tarski  ->  ( A  e.  x  ->  B  C_  x
) ) )
76albii 1575 . . 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 4067 . . 3  |-  ( B 
C_  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) }  <->  A. x ( ( x  e.  Tarski  /\  A  e.  x )  ->  B  C_  x ) )
9 df-ral 2710 . . 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 269 . 2  |-  ( B 
C_  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) }  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  C_  x ) )
115, 10syl6bb 253 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 177    /\ wa 359   A.wal 1549    e. wcel 1725   {cab 2422   A.wral 2705   {crab 2709    C_ wss 3320   |^|cint 4050   ` cfv 5454   Tarskictsk 8623   tarskiMapctskm 8712
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-groth 8698
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-tsk 8624  df-tskm 8713
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