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Theorem fores 5662
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 5492 . . 3  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
21anim1i 552 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( Fun  ( F  |`  A )  /\  A  C_ 
dom  F ) )
3 df-fn 5457 . . 3  |-  ( ( F  |`  A )  Fn  A  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
4 df-ima 4891 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
54eqcomi 2440 . . . 4  |-  ran  ( F  |`  A )  =  ( F " A
)
6 df-fo 5460 . . . 4  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  =  ( F " A
) ) )
75, 6mpbiran2 886 . . 3  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( F  |`  A )  Fn  A
)
8 ssdmres 5168 . . . 4  |-  ( A 
C_  dom  F  <->  dom  ( F  |`  A )  =  A )
98anbi2i 676 . . 3  |-  ( ( Fun  ( F  |`  A )  /\  A  C_ 
dom  F )  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
103, 7, 93bitr4i 269 . 2  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( Fun  ( F  |`  A )  /\  A  C_  dom  F ) )
112, 10sylibr 204 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A ) : A -onto-> ( F
" A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    C_ wss 3320   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881   Fun wfun 5448    Fn wfn 5449   -onto->wfo 5452
This theorem is referenced by:  resdif  5696  f1oweALT  6074  imafi  7399  f1opwfi  7410  fodomfi2  7941  fin1a2lem7  8286  znnen  12812  conima  17488  1stcfb  17508  1stckgenlem  17585  qtoprest  17749  re2ndc  18832  uniiccdif  19470  opnmblALT  19495  mbfimaopnlem  19547  ghsubgolem  21958  erdszelem2  24878  ivthALT  26338  lmhmfgima  27159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-res 4890  df-ima 4891  df-fun 5456  df-fn 5457  df-fo 5460
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