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Theorem isf34lem1 8833
Description: Lemma for isfin3-4 8843. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
compss.a  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
Assertion
Ref Expression
isf34lem1  |-  ( ( A  e.  V  /\  X  C_  A )  -> 
( F `  X
)  =  ( A 
\  X ) )
Distinct variable groups:    x, A    x, V
Allowed substitution hints:    F( x)    X( x)

Proof of Theorem isf34lem1
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 elpw2g 4583 . . 3  |-  ( A  e.  V  ->  ( X  e.  ~P A  <->  X 
C_  A ) )
21biimpar 492 . 2  |-  ( ( A  e.  V  /\  X  C_  A )  ->  X  e.  ~P A
)
3 difexg 4568 . . 3  |-  ( A  e.  V  ->  ( A  \  X )  e. 
_V )
43adantr 471 . 2  |-  ( ( A  e.  V  /\  X  C_  A )  -> 
( A  \  X
)  e.  _V )
5 difeq2 3557 . . 3  |-  ( a  =  X  ->  ( A  \  a )  =  ( A  \  X
) )
6 compss.a . . . 4  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
7 difeq2 3557 . . . . 5  |-  ( x  =  a  ->  ( A  \  x )  =  ( A  \  a
) )
87cbvmptv 4511 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( a  e.  ~P A  |->  ( A  \  a ) )
96, 8eqtri 2484 . . 3  |-  F  =  ( a  e.  ~P A  |->  ( A  \ 
a ) )
105, 9fvmptg 5974 . 2  |-  ( ( X  e.  ~P A  /\  ( A  \  X
)  e.  _V )  ->  ( F `  X
)  =  ( A 
\  X ) )
112, 4, 10syl2anc 671 1  |-  ( ( A  e.  V  /\  X  C_  A )  -> 
( F `  X
)  =  ( A 
\  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   _Vcvv 3057    \ cdif 3413    C_ wss 3416   ~Pcpw 3963    |-> cmpt 4477   ` cfv 5605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-iota 5569  df-fun 5607  df-fv 5613
This theorem is referenced by:  compssiso  8835  isf34lem4  8838  isf34lem7  8840  isf34lem6  8841
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