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Theorem isf34lem2 8640
Description: Lemma for isfin3-4 8649. (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 3578 . . . 4  |-  ( A 
\  x )  C_  A
2 elpw2g 4550 . . . 4  |-  ( A  e.  V  ->  (
( A  \  x
)  e.  ~P A  <->  ( A  \  x ) 
C_  A ) )
31, 2mpbiri 233 . . 3  |-  ( A  e.  V  ->  ( A  \  x )  e. 
~P A )
43adantr 465 . 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 5963 1  |-  ( A  e.  V  ->  F : ~P A --> ~P A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    \ cdif 3420    C_ wss 3423   ~Pcpw 3955    |-> cmpt 4445   -->wf 5509
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521
This theorem is referenced by:  isf34lem5  8645  isf34lem7  8646  isf34lem6  8647
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