Theorem fores 5447

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 5257	. . 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 5219	. . 3 |- ( ( F |` A ) Fn A <-> ( Fun ( F |` A ) /\ dom ( F |` A ) = A ) )
4		df-ima 4639	. . . . 5 |- ( F " A ) = ran ( F |` A )
5	4	eqcomi 2181	. . . 4 |- ran ( F |` A ) = ( F " A )
6		df-fo 5222	. . . 4 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( ( F |` A ) Fn A /\ ran ( F |` A ) = ( F " A ) ) )
7	5, 6	mpbiran2 941	. . 3 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( F |` A ) Fn A )
8		ssdmres 4929	. . . 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 1353   C_ wss 3129   dom cdm 4626   ran crn 4627   |` cres 4628   "cima 4629   Fun wfun 5210   Fn wfn 5211   -onto->wfo 5214

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-res 4638  df-ima 4639  df-fun 5218  df-fn 5219  df-fo 5222

This theorem is referenced by:  resdif  5483  ctinf  12430  qnnen  12431