Theorem dfimafnf 5928

Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)

Hypotheses

Ref	Expression
dfimafnf.1	|- F/_ x A
dfimafnf.2	|- F/_ x F

Assertion

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

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

Allowed substitution hints:   A( x)   F( x)

Proof of Theorem dfimafnf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfima2 5108	. . 3 |- ( F " A ) = { y | E. z e. A z F y }
2		ssel 3236	. . . . . . 7 |- ( A C_ dom F -> ( z e. A -> z e. dom F ) )
3		eqcom 2236	. . . . . . . . 9 |- ( ( F ` z ) = y <-> y = ( F ` z ) )
4		funbrfvb 5722	. . . . . . . . 9 |- ( ( Fun F /\ z e. dom F ) -> ( ( F ` z ) = y <-> z F y ) )
5	3, 4	bitr3id 194	. . . . . . . 8 |- ( ( Fun F /\ z e. dom F ) -> ( y = ( F ` z ) <-> z F y ) )
6	5	ex 115	. . . . . . 7 |- ( Fun F -> ( z e. dom F -> ( y = ( F ` z ) <-> z F y ) ) )
7	2, 6	syl9r 73	. . . . . 6 |- ( Fun F -> ( A C_ dom F -> ( z e. A -> ( y = ( F ` z ) <-> z F y ) ) ) )
8	7	imp31 256	. . . . 5 |- ( ( ( Fun F /\ A C_ dom F ) /\ z e. A ) -> ( y = ( F ` z ) <-> z F y ) )
9	8	rexbidva 2541	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( E. z e. A y = ( F ` z ) <-> E. z e. A z F y ) )
10	9	abbidv 2354	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> { y | E. z e. A y = ( F ` z ) } = { y | E. z e. A z F y } )
11	1, 10	eqtr4id 2286	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. z e. A y = ( F ` z ) } )
12		nfcv 2386	. . . 4 |- F/_ z A
13		dfimafnf.1	. . . 4 |- F/_ x A
14		dfimafnf.2	. . . . . 6 |- F/_ x F
15		nfcv 2386	. . . . . 6 |- F/_ x z
16	14, 15	nffv 5685	. . . . 5 |- F/_ x ( F ` z )
17	16	nfeq2 2398	. . . 4 |- F/ x y = ( F ` z )
18		nfv 1577	. . . 4 |- F/ z y = ( F ` x )
19		fveq2 5675	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
20	19	eqeq2d 2246	. . . 4 |- ( z = x -> ( y = ( F ` z ) <-> y = ( F ` x ) ) )
21	12, 13, 17, 18, 20	cbvrexfw 2770	. . 3 |- ( E. z e. A y = ( F ` z ) <-> E. x e. A y = ( F ` x ) )
22	21	abbii 2350	. 2 |- { y | E. z e. A y = ( F ` z ) } = { y | E. x e. A y = ( F ` x ) }
23	11, 22	eqtrdi 2283	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A y = ( F ` x ) } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {cab 2220   F/_wnfc 2373   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:  funimass4f  6332