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Theorem isf34lem1 8014
Description: Lemma for isfin3-4 8024. (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 4190 . . 3  |-  ( A  e.  V  ->  ( X  e.  ~P A  <->  X 
C_  A ) )
21biimpar 471 . 2  |-  ( ( A  e.  V  /\  X  C_  A )  ->  X  e.  ~P A
)
3 difexg 4178 . . 3  |-  ( A  e.  V  ->  ( A  \  X )  e. 
_V )
43adantr 451 . 2  |-  ( ( A  e.  V  /\  X  C_  A )  -> 
( A  \  X
)  e.  _V )
5 difeq2 3301 . . 3  |-  ( a  =  X  ->  ( A  \  a )  =  ( A  \  X
) )
6 compss.a . . . 4  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
7 difeq2 3301 . . . . 5  |-  ( x  =  a  ->  ( A  \  x )  =  ( A  \  a
) )
87cbvmptv 4127 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( a  e.  ~P A  |->  ( A  \  a ) )
96, 8eqtri 2316 . . 3  |-  F  =  ( a  e.  ~P A  |->  ( A  \ 
a ) )
105, 9fvmptg 5616 . 2  |-  ( ( X  e.  ~P A  /\  ( A  \  X
)  e.  _V )  ->  ( F `  X
)  =  ( A 
\  X ) )
112, 4, 10syl2anc 642 1  |-  ( ( A  e.  V  /\  X  C_  A )  -> 
( F `  X
)  =  ( A 
\  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    C_ wss 3165   ~Pcpw 3638    e. cmpt 4093   ` cfv 5271
This theorem is referenced by:  compssiso  8016  isf34lem4  8019  isf34lem7  8021  isf34lem6  8022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279
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