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Theorem isf34lem1 8216
Description: Lemma for isfin3-4 8226. (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 4331 . . 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 4319 . . 3  |-  ( A  e.  V  ->  ( A  \  X )  e. 
_V )
43adantr 452 . 2  |-  ( ( A  e.  V  /\  X  C_  A )  -> 
( A  \  X
)  e.  _V )
5 difeq2 3427 . . 3  |-  ( a  =  X  ->  ( A  \  a )  =  ( A  \  X
) )
6 compss.a . . . 4  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
7 difeq2 3427 . . . . 5  |-  ( x  =  a  ->  ( A  \  x )  =  ( A  \  a
) )
87cbvmptv 4268 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( a  e.  ~P A  |->  ( A  \  a ) )
96, 8eqtri 2432 . . 3  |-  F  =  ( a  e.  ~P A  |->  ( A  \ 
a ) )
105, 9fvmptg 5771 . 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 1721   _Vcvv 2924    \ cdif 3285    C_ wss 3288   ~Pcpw 3767    e. cmpt 4234   ` cfv 5421
This theorem is referenced by:  compssiso  8218  isf34lem4  8221  isf34lem7  8223  isf34lem6  8224
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429
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