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Theorem isf34lem1 8185
Description: Lemma for isfin3-4 8195. (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 4304 . . 3  |-  ( A  e.  V  ->  ( X  e.  ~P A  <->  X 
C_  A ) )
21biimpar 472 . 2  |-  ( ( A  e.  V  /\  X  C_  A )  ->  X  e.  ~P A
)
3 difexg 4292 . . 3  |-  ( A  e.  V  ->  ( A  \  X )  e. 
_V )
43adantr 452 . 2  |-  ( ( A  e.  V  /\  X  C_  A )  -> 
( A  \  X
)  e.  _V )
5 difeq2 3402 . . 3  |-  ( a  =  X  ->  ( A  \  a )  =  ( A  \  X
) )
6 compss.a . . . 4  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
7 difeq2 3402 . . . . 5  |-  ( x  =  a  ->  ( A  \  x )  =  ( A  \  a
) )
87cbvmptv 4241 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( a  e.  ~P A  |->  ( A  \  a ) )
96, 8eqtri 2407 . . 3  |-  F  =  ( a  e.  ~P A  |->  ( A  \ 
a ) )
105, 9fvmptg 5743 . 2  |-  ( ( X  e.  ~P A  /\  ( A  \  X
)  e.  _V )  ->  ( F `  X
)  =  ( A 
\  X ) )
112, 4, 10syl2anc 643 1  |-  ( ( A  e.  V  /\  X  C_  A )  -> 
( F `  X
)  =  ( A 
\  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899    \ cdif 3260    C_ wss 3263   ~Pcpw 3742    e. cmpt 4207   ` cfv 5394
This theorem is referenced by:  compssiso  8187  isf34lem4  8190  isf34lem7  8192  isf34lem6  8193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402
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