Theorem dfaimafn 41245

Description: Alternate definition of the image of a function, analogous to dfimafn 6245. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
dfaimafn	|- ( ( 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 dfaimafn

Step	Hyp	Ref	Expression
1		ssel 3597	. . . . . 6 |- ( A C_ dom F -> ( x e. A -> x e. dom F ) )
2		funbrafvb 41236	. . . . . . 7 |- ( ( Fun F /\ x e. dom F ) -> ( ( F''' x ) = y <-> x F y ) )
3	2	ex 450	. . . . . 6 |- ( Fun F -> ( x e. dom F -> ( ( F''' x ) = y <-> x F y ) ) )
4	1, 3	syl9r 78	. . . . 5 |- ( Fun F -> ( A C_ dom F -> ( x e. A -> ( ( F''' x ) = y <-> x F y ) ) ) )
5	4	imp31 448	. . . 4 |- ( ( ( Fun F /\ A C_ dom F ) /\ x e. A ) -> ( ( F''' x ) = y <-> x F y ) )
6	5	rexbidva 3049	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( E. x e. A ( F''' x ) = y <-> E. x e. A x F y ) )
7	6	abbidv 2741	. 2 |- ( ( Fun F /\ A C_ dom F ) -> { y | E. x e. A ( F''' x ) = y } = { y | E. x e. A x F y } )
8		dfima2 5468	. 2 |- ( F " A ) = { y | E. x e. A x F y }
9	7, 8	syl6reqr 2675	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A ( F''' x ) = y } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   C_ wss 3574   class class class wbr 4653   dom cdm 5114   "cima 5117   Fun wfun 5882  '''cafv 41194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by:  dfaimafn2  41246