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Theorem isf34lem2 8742
Description: Lemma for isfin3-4 8751. (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 3624 . . . 4  |-  ( A 
\  x )  C_  A
2 elpw2g 4603 . . . 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 6036 1  |-  ( A  e.  V  ->  F : ~P A --> ~P A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    \ cdif 3466    C_ wss 3469   ~Pcpw 4003    |-> cmpt 4498   -->wf 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587
This theorem is referenced by:  isf34lem5  8747  isf34lem7  8748  isf34lem6  8749
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