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Theorem fores 5819
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 5640 . . 3  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
21anim1i 570 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( Fun  ( F  |`  A )  /\  A  C_ 
dom  F ) )
3 df-fn 5604 . . 3  |-  ( ( F  |`  A )  Fn  A  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
4 df-ima 4867 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
54eqcomi 2442 . . . 4  |-  ran  ( F  |`  A )  =  ( F " A
)
6 df-fo 5607 . . . 4  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  =  ( F " A
) ) )
75, 6mpbiran2 927 . . 3  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( F  |`  A )  Fn  A
)
8 ssdmres 5146 . . . 4  |-  ( A 
C_  dom  F  <->  dom  ( F  |`  A )  =  A )
98anbi2i 698 . . 3  |-  ( ( Fun  ( F  |`  A )  /\  A  C_ 
dom  F )  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
103, 7, 93bitr4i 280 . 2  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( Fun  ( F  |`  A )  /\  A  C_  dom  F ) )
112, 10sylibr 215 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A ) : A -onto-> ( F
" A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    C_ wss 3442   dom cdm 4854   ran crn 4855    |` cres 4856   "cima 4857   Fun wfun 5595    Fn wfn 5596   -onto->wfo 5599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-ima 4867  df-fun 5603  df-fn 5604  df-fo 5607
This theorem is referenced by:  resdif  5851  f1oweALT  6791  imafi  7873  f1opwfi  7884  fodomfi2  8489  fin1a2lem7  8834  znnen  14243  conima  20371  1stcfb  20391  1stckgenlem  20499  qtoprest  20663  re2ndc  21730  uniiccdif  22412  opnmblALT  22438  mbfimaopnlem  22488  ghsubgolemOLD  25943  ffsrn  28157  erdszelem2  29703  ivthALT  30776  poimirlem26  31670  poimirlem27  31671  lmhmfgima  35648
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