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Theorem isf34lem3 8751
Description: Lemma for isfin3-4 8758. (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 8747 . . 3  |-  `' F  =  F
32imaeq1i 5332 . 2  |-  ( `' F " ( F
" X ) )  =  ( F "
( F " X
) )
41compssiso 8750 . . . 4  |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A ) )
5 isof1o 6207 . . . 4  |-  ( F 
Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A )  ->  F : ~P A -1-1-onto-> ~P A )
6 f1of1 5813 . . . 4  |-  ( F : ~P A -1-1-onto-> ~P A  ->  F : ~P A -1-1-> ~P A )
74, 5, 63syl 20 . . 3  |-  ( A  e.  V  ->  F : ~P A -1-1-> ~P A
)
8 f1imacnv 5830 . . 3  |-  ( ( F : ~P A -1-1-> ~P A  /\  X  C_  ~P A )  ->  ( `' F " ( F
" X ) )  =  X )
97, 8sylan 471 . 2  |-  ( ( A  e.  V  /\  X  C_  ~P A )  ->  ( `' F " ( F " X
) )  =  X )
103, 9syl5eqr 2522 1  |-  ( ( A  e.  V  /\  X  C_  ~P A )  ->  ( F "
( F " X
) )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3473    C_ wss 3476   ~Pcpw 4010    |-> cmpt 4505   `'ccnv 4998   "cima 5002   -1-1->wf1 5583   -1-1-onto->wf1o 5585    Isom wiso 5587   [ C.] crpss 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-rpss 6562
This theorem is referenced by:  isf34lem5  8754  isf34lem7  8755  isf34lem6  8756
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