Theorem fores 5558

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 5359	. . 3 |- ( Fun F -> Fun ( F |` A ) )
2	1	anim1i 340	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( Fun ( F |` A ) /\ A C_ dom F ) )
3		df-fn 5321	. . 3 |- ( ( F |` A ) Fn A <-> ( Fun ( F |` A ) /\ dom ( F |` A ) = A ) )
4		df-ima 4732	. . . . 5 |- ( F " A ) = ran ( F |` A )
5	4	eqcomi 2233	. . . 4 |- ran ( F |` A ) = ( F " A )
6		df-fo 5324	. . . 4 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( ( F |` A ) Fn A /\ ran ( F |` A ) = ( F " A ) ) )
7	5, 6	mpbiran2 947	. . 3 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( F |` A ) Fn A )
8		ssdmres 5027	. . . 4 |- ( A C_ dom F <-> dom ( F |` A ) = A )
9	8	anbi2i 457	. . 3 |- ( ( Fun ( F |` A ) /\ A C_ dom F ) <-> ( Fun ( F |` A ) /\ dom ( F |` A ) = A ) )
10	3, 7, 9	3bitr4i 212	. 2 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( Fun ( F |` A ) /\ A C_ dom F ) )
11	2, 10	sylibr 134	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   C_ wss 3197   dom cdm 4719   ran crn 4720   |` cres 4721   "cima 4722   Fun wfun 5312   Fn wfn 5313   -onto->wfo 5316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-res 4731  df-ima 4732  df-fun 5320  df-fn 5321  df-fo 5324

This theorem is referenced by:  resdif  5594  ctinf  13001  qnnen  13002