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Theorem sstskm 9209
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 9206 . . . 4  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
2 df-rab 2813 . . . . 5  |-  { x  e.  Tarski  |  A  e.  x }  =  {
x  |  ( x  e.  Tarski  /\  A  e.  x ) }
32inteqi 4275 . . . 4  |-  |^| { x  e.  Tarski  |  A  e.  x }  =  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) }
41, 3syl6eq 2511 . . 3  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) } )
54sseq2d 3517 . 2  |-  ( A  e.  V  ->  ( B  C_  ( tarskiMap `  A
)  <->  B  C_  |^| { x  |  ( x  e. 
Tarski  /\  A  e.  x
) } ) )
6 impexp 444 . . . 4  |-  ( ( ( x  e.  Tarski  /\  A  e.  x )  ->  B  C_  x
)  <->  ( x  e. 
Tarski  ->  ( A  e.  x  ->  B  C_  x
) ) )
76albii 1645 . . 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 4288 . . 3  |-  ( B 
C_  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) }  <->  A. x ( ( x  e.  Tarski  /\  A  e.  x )  ->  B  C_  x ) )
9 df-ral 2809 . . 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 277 . 2  |-  ( B 
C_  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) }  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  C_  x ) )
115, 10syl6bb 261 1  |-  ( A  e.  V  ->  ( B  C_  ( tarskiMap `  A
)  <->  A. x  e.  Tarski  ( A  e.  x  ->  B  C_  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396    e. wcel 1823   {cab 2439   A.wral 2804   {crab 2808    C_ wss 3461   |^|cint 4271   ` cfv 5570   Tarskictsk 9115   tarskiMapctskm 9204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-groth 9190
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-tsk 9116  df-tskm 9205
This theorem is referenced by: (None)
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