Theorem dfimafn 5730

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 5108	. 2 |- ( F " A ) = { y | E. x e. A x F y }
2		ssel 3236	. . . . . 6 |- ( A C_ dom F -> ( x e. A -> x e. dom F ) )
3		funbrfvb 5722	. . . . . . 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 2541	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( E. x e. A ( F ` x ) = y <-> E. x e. A x F y ) )
8	7	abbidv 2354	. 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 2286	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 1398   e. wcel 2205   {cab 2220   E.wrex 2523   C_ wss 3214   class class class wbr 4114   dom cdm 4754   "cima 4757   Fun wfun 5351   ` cfv 5357

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365

This theorem is referenced by:  dfimafn2  5731  fvelimab  5738  ushgredgedg  16347  ushgredgedgloop  16349