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Theorem fores 5242
Description: Restriction of a function. (Contributed by NM, 4-Mar-1997.)
Assertion
Ref Expression
fores  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A ) : A -onto-> ( F
" A ) )

Proof of Theorem fores
StepHypRef Expression
1 funres 5055 . . 3  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
21anim1i 333 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( Fun  ( F  |`  A )  /\  A  C_ 
dom  F ) )
3 df-fn 5018 . . 3  |-  ( ( F  |`  A )  Fn  A  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
4 df-ima 4451 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
54eqcomi 2092 . . . 4  |-  ran  ( F  |`  A )  =  ( F " A
)
6 df-fo 5021 . . . 4  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  =  ( F " A
) ) )
75, 6mpbiran2 887 . . 3  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( F  |`  A )  Fn  A
)
8 ssdmres 4735 . . . 4  |-  ( A 
C_  dom  F  <->  dom  ( F  |`  A )  =  A )
98anbi2i 445 . . 3  |-  ( ( Fun  ( F  |`  A )  /\  A  C_ 
dom  F )  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
103, 7, 93bitr4i 210 . 2  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( Fun  ( F  |`  A )  /\  A  C_  dom  F ) )
112, 10sylibr 132 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A ) : A -onto-> ( F
" A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    C_ wss 2999   dom cdm 4438   ran crn 4439    |` cres 4440   "cima 4441   Fun wfun 5009    Fn wfn 5010   -onto->wfo 5013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-res 4450  df-ima 4451  df-fun 5017  df-fn 5018  df-fo 5021
This theorem is referenced by:  resdif  5275
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