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Theorem isf34lem2 8834
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
isf34lem2  |-  ( A  e.  V  ->  F : ~P A --> ~P A
)
Distinct variable groups:    x, A    x, V
Allowed substitution hint:    F( x)

Proof of Theorem isf34lem2
StepHypRef Expression
1 difss 3572 . . . 4  |-  ( A 
\  x )  C_  A
2 elpw2g 4583 . . . 4  |-  ( A  e.  V  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
31, 2mpbiri 241 . . 3  |-  ( A  e.  V  ->  ( A  \  x )  e. 
~P A )
43adantr 471 . 2  |-  ( ( A  e.  V  /\  x  e.  ~P A
)  ->  ( A  \  x )  e.  ~P A )
5 compss.a . 2  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
64, 5fmptd 6074 1  |-  ( A  e.  V  ->  F : ~P A --> ~P A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1455    e. wcel 1898    \ cdif 3413    C_ wss 3416   ~Pcpw 3963    |-> cmpt 4477   -->wf 5601
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-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-fv 5613
This theorem is referenced by:  isf34lem5  8839  isf34lem7  8840  isf34lem6  8841
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