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Theorem isf34lem3 8001
Description: Lemma for isfin3-4 8008. (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 7997 . . 3  |-  `' F  =  F
32imaeq1i 5009 . 2  |-  ( `' F " ( F
" X ) )  =  ( F "
( F " X
) )
41compssiso 8000 . . . 4  |-  ( A  e.  V  ->  F  Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A ) )
5 isof1o 5822 . . . 4  |-  ( F 
Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A )  ->  F : ~P A -1-1-onto-> ~P A )
6 f1of1 5471 . . . 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 5489 . . 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 2329 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 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   ~Pcpw 3625    e. cmpt 4077   `'ccnv 4688   "cima 4692   -1-1->wf1 5252   -1-1-onto->wf1o 5254    Isom wiso 5256   [ C.] crpss 6276
This theorem is referenced by:  isf34lem5  8004  isf34lem7  8005  isf34lem6  8006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-rpss 6277
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