Theorem fores 5135

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

Step	Hyp	Ref	Expression
1		funres 4961	. . 3 |- ( Fun F -> Fun ( F |` A ) )
2	1	anim1i 333	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( Fun ( F |` A ) /\ A C_ dom F ) )
3		df-fn 4925	. . 3 |- ( ( F |` A ) Fn A <-> ( Fun ( F |` A ) /\ dom ( F |` A ) = A ) )
4		df-ima 4376	. . . . 5 |- ( F " A ) = ran ( F |` A )
5	4	eqcomi 2085	. . . 4 |- ran ( F |` A ) = ( F " A )
6		df-fo 4928	. . . 4 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( ( F |` A ) Fn A /\ ran ( F |` A ) = ( F " A ) ) )
7	5, 6	mpbiran2 882	. . 3 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( F |` A ) Fn A )
8		ssdmres 4651	. . . 4 |- ( A C_ dom F <-> dom ( F |` A ) = A )
9	8	anbi2i 444	. . 3 |- ( ( Fun ( F |` A ) /\ A C_ dom F ) <-> ( Fun ( F |` A ) /\ dom ( F |` A ) = A ) )
10	3, 7, 9	3bitr4i 210	. 2 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( Fun ( F |` A ) /\ A C_ dom F ) )
11	2, 10	sylibr 132	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   C_ wss 2973   dom cdm 4363   ran crn 4364   |` cres 4365   "cima 4366   Fun wfun 4916   Fn wfn 4917   -onto->wfo 4920

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-res 4375  df-ima 4376  df-fun 4924  df-fn 4925  df-fo 4928

This theorem is referenced by:  resdif  5168