Theorem dfimafn 5690

Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
dfimafn	|- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A ( F ` x ) = y } )

Distinct variable groups:   x, y, A   x, F, y

Proof of Theorem dfimafn

Step	Hyp	Ref	Expression
1		dfima2 5076	. 2 |- ( F " A ) = { y | E. x e. A x F y }
2		ssel 3219	. . . . . 6 |- ( A C_ dom F -> ( x e. A -> x e. dom F ) )
3		funbrfvb 5682	. . . . . . 7 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) = y <-> x F y ) )
4	3	ex 115	. . . . . 6 |- ( Fun F -> ( x e. dom F -> ( ( F ` x ) = y <-> x F y ) ) )
5	2, 4	syl9r 73	. . . . 5 |- ( Fun F -> ( A C_ dom F -> ( x e. A -> ( ( F ` x ) = y <-> x F y ) ) ) )
6	5	imp31 256	. . . 4 |- ( ( ( Fun F /\ A C_ dom F ) /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) )
7	6	rexbidva 2527	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( E. x e. A ( F ` x ) = y <-> E. x e. A x F y ) )
8	7	abbidv 2347	. 2 |- ( ( Fun F /\ A C_ dom F ) -> { y | E. x e. A ( F ` x ) = y } = { y | E. x e. A x F y } )
9	1, 8	eqtr4id 2281	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A ( F ` x ) = y } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   C_ wss 3198   class class class wbr 4086   dom cdm 4723   "cima 4726   Fun wfun 5318   ` cfv 5324

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 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332

This theorem is referenced by:  dfimafn2  5691  fvelimab  5698  ushgredgedg  16065  ushgredgedgloop  16067