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Theorem isf34lem2 8044
Description: Lemma for isfin3-4 8053. (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 3337 . . . 4  |-  ( A 
\  x )  C_  A
2 elpw2g 4211 . . . 4  |-  ( A  e.  V  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
31, 2mpbiri 224 . . 3  |-  ( A  e.  V  ->  ( A  \  x )  e. 
~P A )
43adantr 451 . 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 5722 1  |-  ( A  e.  V  ->  F : ~P A --> ~P A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701    \ cdif 3183    C_ wss 3186   ~Pcpw 3659    e. cmpt 4114   -->wf 5288
This theorem is referenced by:  isf34lem5  8049  isf34lem7  8050  isf34lem6  8051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300
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