Theorem fores 5419

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 5229	. . 3 |- ( Fun F -> Fun ( F |` A ) )
2	1	anim1i 338	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( Fun ( F |` A ) /\ A C_ dom F ) )
3		df-fn 5191	. . 3 |- ( ( F |` A ) Fn A <-> ( Fun ( F |` A ) /\ dom ( F |` A ) = A ) )
4		df-ima 4617	. . . . 5 |- ( F " A ) = ran ( F |` A )
5	4	eqcomi 2169	. . . 4 |- ran ( F |` A ) = ( F " A )
6		df-fo 5194	. . . 4 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( ( F |` A ) Fn A /\ ran ( F |` A ) = ( F " A ) ) )
7	5, 6	mpbiran2 931	. . 3 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( F |` A ) Fn A )
8		ssdmres 4906	. . . 4 |- ( A C_ dom F <-> dom ( F |` A ) = A )
9	8	anbi2i 453	. . 3 |- ( ( Fun ( F |` A ) /\ A C_ dom F ) <-> ( Fun ( F |` A ) /\ dom ( F |` A ) = A ) )
10	3, 7, 9	3bitr4i 211	. 2 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( Fun ( F |` A ) /\ A C_ dom F ) )
11	2, 10	sylibr 133	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F |` A ) : A -onto-> ( F " A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   C_ wss 3116   dom cdm 4604   ran crn 4605   |` cres 4606   "cima 4607   Fun wfun 5182   Fn wfn 5183   -onto->wfo 5186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-res 4616  df-ima 4617  df-fun 5190  df-fn 5191  df-fo 5194

This theorem is referenced by:  resdif  5454  ctinf  12363  qnnen  12364