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Theorem isf34lem1 8253
Description: Lemma for isfin3-4 8263. (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 4364 . . 3  |-  ( A  e.  V  ->  ( X  e.  ~P A  <->  X 
C_  A ) )
21biimpar 473 . 2  |-  ( ( A  e.  V  /\  X  C_  A )  ->  X  e.  ~P A
)
3 difexg 4352 . . 3  |-  ( A  e.  V  ->  ( A  \  X )  e. 
_V )
43adantr 453 . 2  |-  ( ( A  e.  V  /\  X  C_  A )  -> 
( A  \  X
)  e.  _V )
5 difeq2 3460 . . 3  |-  ( a  =  X  ->  ( A  \  a )  =  ( A  \  X
) )
6 compss.a . . . 4  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
7 difeq2 3460 . . . . 5  |-  ( x  =  a  ->  ( A  \  x )  =  ( A  \  a
) )
87cbvmptv 4301 . . . 4  |-  ( x  e.  ~P A  |->  ( A  \  x ) )  =  ( a  e.  ~P A  |->  ( A  \  a ) )
96, 8eqtri 2457 . . 3  |-  F  =  ( a  e.  ~P A  |->  ( A  \ 
a ) )
105, 9fvmptg 5805 . 2  |-  ( ( X  e.  ~P A  /\  ( A  \  X
)  e.  _V )  ->  ( F `  X
)  =  ( A 
\  X ) )
112, 4, 10syl2anc 644 1  |-  ( ( A  e.  V  /\  X  C_  A )  -> 
( F `  X
)  =  ( A 
\  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2957    \ cdif 3318    C_ wss 3321   ~Pcpw 3800    e. cmpt 4267   ` cfv 5455
This theorem is referenced by:  compssiso  8255  isf34lem4  8258  isf34lem7  8260  isf34lem6  8261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463
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