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Theorem fores 5460
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 5293 . . 3  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
21anim1i 551 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( Fun  ( F  |`  A )  /\  A  C_ 
dom  F ) )
3 df-fn 5258 . . 3  |-  ( ( F  |`  A )  Fn  A  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
4 df-ima 4702 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
54eqcomi 2287 . . . 4  |-  ran  ( F  |`  A )  =  ( F " A
)
6 df-fo 5261 . . . 4  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  =  ( F " A
) ) )
75, 6mpbiran2 885 . . 3  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( F  |`  A )  Fn  A
)
8 ssdmres 4977 . . . 4  |-  ( A 
C_  dom  F  <->  dom  ( F  |`  A )  =  A )
98anbi2i 675 . . 3  |-  ( ( Fun  ( F  |`  A )  /\  A  C_ 
dom  F )  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
103, 7, 93bitr4i 268 . 2  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( Fun  ( F  |`  A )  /\  A  C_  dom  F ) )
112, 10sylibr 203 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A ) : A -onto-> ( F
" A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    C_ wss 3152   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   -onto->wfo 5253
This theorem is referenced by:  resdif  5494  f1oweALT  5851  imafi  7148  f1opwfi  7159  fodomfi2  7687  fin1a2lem7  8032  znnen  12491  conima  17151  1stcfb  17171  1stckgenlem  17248  qtoprest  17408  re2ndc  18307  uniiccdif  18933  opnmblALT  18958  mbfimaopnlem  19010  ghsubgolem  21037  erdszelem2  23723  ivthALT  26258  lmhmfgima  27182
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-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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-fo 5261
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