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Theorem isf34lem3 8017
Description: Lemma for isfin3-4 8024. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
Assertion
Ref Expression
isf34lem3  |-  ( ( A  e.  V  /\  X  C_  ~P A )  ->  ( F "
( F " X
) )  =  X )
Distinct variable groups:    x, A    x, V
Allowed substitution hints:    F( x)    X( x)

Proof of Theorem isf34lem3
StepHypRef Expression
1 compss.a . . . 4  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
21compsscnv 8013 . . 3  |-  `' F  =  F
32imaeq1i 5025 . 2  |-  ( `' F " ( F
" X ) )  =  ( F "
( F " X
) )
41compssiso 8016 . . . 4  |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A ) )
5 isof1o 5838 . . . 4  |-  ( F 
Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A )  ->  F : ~P A -1-1-onto-> ~P A )
6 f1of1 5487 . . . 4  |-  ( F : ~P A -1-1-onto-> ~P A  ->  F : ~P A -1-1-> ~P A )
74, 5, 63syl 18 . . 3  |-  ( A  e.  V  ->  F : ~P A -1-1-> ~P A
)
8 f1imacnv 5505 . . 3  |-  ( ( F : ~P A -1-1-> ~P A  /\  X  C_  ~P A )  ->  ( `' F " ( F
" X ) )  =  X )
97, 8sylan 457 . 2  |-  ( ( A  e.  V  /\  X  C_  ~P A )  ->  ( `' F " ( F " X
) )  =  X )
103, 9syl5eqr 2342 1  |-  ( ( A  e.  V  /\  X  C_  ~P A )  ->  ( F "
( F " X
) )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162    C_ wss 3165   ~Pcpw 3638    e. cmpt 4093   `'ccnv 4704   "cima 4708   -1-1->wf1 5268   -1-1-onto->wf1o 5270    Isom wiso 5272   [ C.] crpss 6292
This theorem is referenced by:  isf34lem5  8020  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-pss 3181  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-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-rpss 6293
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