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Theorem isf34lem3 8772
Description: Lemma for isfin3-4 8779. (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 8768 . . 3  |-  `' F  =  F
32imaeq1i 5344 . 2  |-  ( `' F " ( F
" X ) )  =  ( F "
( F " X
) )
41compssiso 8771 . . . 4  |-  ( A  e.  V  ->  F  Isom [
C.]  ,  `' [ C.]  ( ~P A ,  ~P A
) )
5 isof1o 6222 . . . 4  |-  ( F 
Isom [ C.]  ,  `' [ C.]  ( ~P A ,  ~P A
)  ->  F : ~P A -1-1-onto-> ~P A )
6 f1of1 5821 . . . 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 5838 . . 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 2512 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 1395    e. wcel 1819    \ cdif 3468    C_ wss 3471   ~Pcpw 4015    |-> cmpt 4515   `'ccnv 5007   "cima 5011   -1-1->wf1 5591   -1-1-onto->wf1o 5593    Isom wiso 5595   [ C.] crpss 6578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-rpss 6579
This theorem is referenced by:  isf34lem5  8775  isf34lem7  8776  isf34lem6  8777
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