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Theorem fores 5628
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 5456 . . 3  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
21anim1i 568 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( Fun  ( F  |`  A )  /\  A  C_ 
dom  F ) )
3 df-fn 5420 . . 3  |-  ( ( F  |`  A )  Fn  A  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
4 df-ima 4852 . . . . 5  |-  ( F
" A )  =  ran  ( F  |`  A )
54eqcomi 2446 . . . 4  |-  ran  ( F  |`  A )  =  ( F " A
)
6 df-fo 5423 . . . 4  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( ( F  |`  A )  Fn  A  /\  ran  ( F  |`  A )  =  ( F " A
) ) )
75, 6mpbiran2 910 . . 3  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( F  |`  A )  Fn  A
)
8 ssdmres 5131 . . . 4  |-  ( A 
C_  dom  F  <->  dom  ( F  |`  A )  =  A )
98anbi2i 694 . . 3  |-  ( ( Fun  ( F  |`  A )  /\  A  C_ 
dom  F )  <->  ( Fun  ( F  |`  A )  /\  dom  ( F  |`  A )  =  A ) )
103, 7, 93bitr4i 277 . 2  |-  ( ( F  |`  A ) : A -onto-> ( F " A )  <->  ( Fun  ( F  |`  A )  /\  A  C_  dom  F ) )
112, 10sylibr 212 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F  |`  A ) : A -onto-> ( F
" A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    C_ wss 3327   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Fun wfun 5411    Fn wfn 5412   -onto->wfo 5415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-br 4292  df-opab 4350  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-res 4851  df-ima 4852  df-fun 5419  df-fn 5420  df-fo 5423
This theorem is referenced by:  resdif  5660  f1oweALT  6560  imafi  7603  f1opwfi  7614  fodomfi2  8229  fin1a2lem7  8574  znnen  13494  conima  19028  1stcfb  19048  1stckgenlem  19125  qtoprest  19289  re2ndc  20377  uniiccdif  21057  opnmblALT  21082  mbfimaopnlem  21132  ghsubgolem  23856  ffsrn  26028  erdszelem2  27079  ivthALT  28528  lmhmfgima  29435
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