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Theorem sstskm 9124
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 9121 . . . 4  |-  ( A  e.  V  ->  ( tarskiMap `  A )  =  |^| { x  e.  Tarski  |  A  e.  x } )
2 df-rab 2808 . . . . 5  |-  { x  e.  Tarski  |  A  e.  x }  =  {
x  |  ( x  e.  Tarski  /\  A  e.  x ) }
32inteqi 4243 . . . 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 3495 . 2  |-  ( A  e.  V  ->  ( B  C_  ( tarskiMap `  A
)  <->  B  C_  |^| { x  |  ( x  e. 
Tarski  /\  A  e.  x
) } ) )
6 impexp 446 . . . 4  |-  ( ( ( x  e.  Tarski  /\  A  e.  x )  ->  B  C_  x
)  <->  ( x  e. 
Tarski  ->  ( A  e.  x  ->  B  C_  x
) ) )
76albii 1611 . . 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 4256 . . 3  |-  ( B 
C_  |^| { x  |  ( x  e.  Tarski  /\  A  e.  x ) }  <->  A. x ( ( x  e.  Tarski  /\  A  e.  x )  ->  B  C_  x ) )
9 df-ral 2804 . . 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 369   A.wal 1368    e. wcel 1758   {cab 2439   A.wral 2799   {crab 2803    C_ wss 3439   |^|cint 4239   ` cfv 5529   Tarskictsk 9030   tarskiMapctskm 9119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-groth 9105
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-tsk 9031  df-tskm 9120
This theorem is referenced by: (None)
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