Theorem fores 5429

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 5239	. . 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 5201	. . 3 |- ( ( F |` A ) Fn A <-> ( Fun ( F |` A ) /\ dom ( F |` A ) = A ) )
4		df-ima 4624	. . . . 5 |- ( F " A ) = ran ( F |` A )
5	4	eqcomi 2174	. . . 4 |- ran ( F |` A ) = ( F " A )
6		df-fo 5204	. . . 4 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( ( F |` A ) Fn A /\ ran ( F |` A ) = ( F " A ) ) )
7	5, 6	mpbiran2 936	. . 3 |- ( ( F |` A ) : A -onto-> ( F " A ) <-> ( F |` A ) Fn A )
8		ssdmres 4913	. . . 4 |- ( A C_ dom F <-> dom ( F |` A ) = A )
9	8	anbi2i 454	. . 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 1348   C_ wss 3121   dom cdm 4611   ran crn 4612   |` cres 4613   "cima 4614   Fun wfun 5192   Fn wfn 5193   -onto->wfo 5196

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-ima 4624  df-fun 5200  df-fn 5201  df-fo 5204

This theorem is referenced by:  resdif  5464  ctinf  12385  qnnen  12386