Theorem dfimafn 5626

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 5023	. 2 |- ( F " A ) = { y | E. x e. A x F y }
2		ssel 3186	. . . . . 6 |- ( A C_ dom F -> ( x e. A -> x e. dom F ) )
3		funbrfvb 5620	. . . . . . 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 2502	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( E. x e. A ( F ` x ) = y <-> E. x e. A x F y ) )
8	7	abbidv 2322	. 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 2256	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 1372   e. wcel 2175   {cab 2190   E.wrex 2484   C_ wss 3165   class class class wbr 4043   dom cdm 4674   "cima 4677   Fun wfun 5264   ` cfv 5270

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278

This theorem is referenced by:  dfimafn2  5627  fvelimab  5634