Theorem dfimafnf 29436

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		ssel 3597	. . . . . . 7 |- ( A C_ dom F -> ( z e. A -> z e. dom F ) )
2		eqcom 2629	. . . . . . . . 9 |- ( ( F ` z ) = y <-> y = ( F ` z ) )
3		funbrfvb 6238	. . . . . . . . 9 |- ( ( Fun F /\ z e. dom F ) -> ( ( F ` z ) = y <-> z F y ) )
4	2, 3	syl5bbr 274	. . . . . . . 8 |- ( ( Fun F /\ z e. dom F ) -> ( y = ( F ` z ) <-> z F y ) )
5	4	ex 450	. . . . . . 7 |- ( Fun F -> ( z e. dom F -> ( y = ( F ` z ) <-> z F y ) ) )
6	1, 5	syl9r 78	. . . . . 6 |- ( Fun F -> ( A C_ dom F -> ( z e. A -> ( y = ( F ` z ) <-> z F y ) ) ) )
7	6	imp31 448	. . . . 5 |- ( ( ( Fun F /\ A C_ dom F ) /\ z e. A ) -> ( y = ( F ` z ) <-> z F y ) )
8	7	rexbidva 3049	. . . 4 |- ( ( Fun F /\ A C_ dom F ) -> ( E. z e. A y = ( F ` z ) <-> E. z e. A z F y ) )
9	8	abbidv 2741	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> { y | E. z e. A y = ( F ` z ) } = { y | E. z e. A z F y } )
10		dfima2 5468	. . 3 |- ( F " A ) = { y | E. z e. A z F y }
11	9, 10	syl6reqr 2675	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. z e. A y = ( F ` z ) } )
12		nfcv 2764	. . . 4 |- F/_ z A
13		dfimafnf.1	. . . 4 |- F/_ x A
14		dfimafnf.2	. . . . . 6 |- F/_ x F
15		nfcv 2764	. . . . . 6 |- F/_ x z
16	14, 15	nffv 6198	. . . . 5 |- F/_ x ( F ` z )
17	16	nfeq2 2780	. . . 4 |- F/ x y = ( F ` z )
18		nfv 1843	. . . 4 |- F/ z y = ( F ` x )
19		fveq2 6191	. . . . 5 |- ( z = x -> ( F ` z ) = ( F ` x ) )
20	19	eqeq2d 2632	. . . 4 |- ( z = x -> ( y = ( F ` z ) <-> y = ( F ` x ) ) )
21	12, 13, 17, 18, 20	cbvrexf 3166	. . 3 |- ( E. z e. A y = ( F ` z ) <-> E. x e. A y = ( F ` x ) )
22	21	abbii 2739	. 2 |- { y | E. z e. A y = ( F ` z ) } = { y | E. x e. A y = ( F ` x ) }
23	11, 22	syl6eq 2672	1 |- ( ( Fun F /\ A C_ dom F ) -> ( F " A ) = { y | E. x e. A y = ( F ` x ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   E.wrex 2913   C_ wss 3574   class class class wbr 4653   dom cdm 5114   "cima 5117   Fun wfun 5882   ` cfv 5888

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

This theorem is referenced by:  funimass4f  29437