Theorem imageval 32037

Description: The image functor in maps-to notation. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
imageval	|- Image R = ( x e. _V |-> ( R " x ) )

Distinct variable group:   x, R

Proof of Theorem imageval

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimage 32035	. . 3 |- Fun Image R
2		funrel 5905	. . 3 |- ( Fun Image R -> Rel Image R )
3	1, 2	ax-mp 5	. 2 |- Rel Image R
4		mptrel 5248	. 2 |- Rel ( x e. _V |-> ( R " x ) )
5		vex 3203	. . . . 5 |- y e. _V
6		vex 3203	. . . . 5 |- z e. _V
7	5, 6	breldm 5329	. . . 4 |- ( yImage R z -> y e. dom Image R )
8		fnimage 32036	. . . . 5 |- Image R Fn { x | ( R " x ) e. _V }
9		fndm 5990	. . . . 5 |- (Image R Fn { x | ( R " x ) e. _V } -> dom Image R = { x | ( R " x ) e. _V } )
10	8, 9	ax-mp 5	. . . 4 |- dom Image R = { x | ( R " x ) e. _V }
11	7, 10	syl6eleq 2711	. . 3 |- ( yImage R z -> y e. { x | ( R " x ) e. _V } )
12	5, 6	breldm 5329	. . . 4 |- ( y ( x e. _V |-> ( R " x ) ) z -> y e. dom ( x e. _V |-> ( R " x ) ) )
13		eqid 2622	. . . . . 6 |- ( x e. _V |-> ( R " x ) ) = ( x e. _V |-> ( R " x ) )
14	13	dmmpt 5630	. . . . 5 |- dom ( x e. _V |-> ( R " x ) ) = { x e. _V | ( R " x ) e. _V }
15		rabab 3223	. . . . 5 |- { x e. _V | ( R " x ) e. _V } = { x | ( R " x ) e. _V }
16	14, 15	eqtri 2644	. . . 4 |- dom ( x e. _V |-> ( R " x ) ) = { x | ( R " x ) e. _V }
17	12, 16	syl6eleq 2711	. . 3 |- ( y ( x e. _V |-> ( R " x ) ) z -> y e. { x | ( R " x ) e. _V } )
18		imaeq2 5462	. . . . . 6 |- ( x = y -> ( R " x ) = ( R " y ) )
19	18	eleq1d 2686	. . . . 5 |- ( x = y -> ( ( R " x ) e. _V <-> ( R " y ) e. _V ) )
20	5, 19	elab 3350	. . . 4 |- ( y e. { x | ( R " x ) e. _V } <-> ( R " y ) e. _V )
21	5, 6	brimage 32033	. . . . 5 |- ( yImage R z <-> z = ( R " y ) )
22		eqcom 2629	. . . . . 6 |- ( z = ( R " y ) <-> ( R " y ) = z )
23	18, 13	fvmptg 6280	. . . . . . . . 9 |- ( ( y e. _V /\ ( R " y ) e. _V ) -> ( ( x e. _V |-> ( R " x ) ) ` y ) = ( R " y ) )
24	5, 23	mpan 706	. . . . . . . 8 |- ( ( R " y ) e. _V -> ( ( x e. _V |-> ( R " x ) ) ` y ) = ( R " y ) )
25	24	eqeq1d 2624	. . . . . . 7 |- ( ( R " y ) e. _V -> ( ( ( x e. _V |-> ( R " x ) ) ` y ) = z <-> ( R " y ) = z ) )
26		funmpt 5926	. . . . . . . . 9 |- Fun ( x e. _V |-> ( R " x ) )
27		df-fn 5891	. . . . . . . . 9 |- ( ( x e. _V |-> ( R " x ) ) Fn { x | ( R " x ) e. _V } <-> ( Fun ( x e. _V |-> ( R " x ) ) /\ dom ( x e. _V |-> ( R " x ) ) = { x | ( R " x ) e. _V } ) )
28	26, 16, 27	mpbir2an 955	. . . . . . . 8 |- ( x e. _V |-> ( R " x ) ) Fn { x | ( R " x ) e. _V }
29	20	biimpri 218	. . . . . . . 8 |- ( ( R " y ) e. _V -> y e. { x | ( R " x ) e. _V } )
30		fnbrfvb 6236	. . . . . . . 8 |- ( ( ( x e. _V |-> ( R " x ) ) Fn { x | ( R " x ) e. _V } /\ y e. { x | ( R " x ) e. _V } ) -> ( ( ( x e. _V |-> ( R " x ) ) ` y ) = z <-> y ( x e. _V |-> ( R " x ) ) z ) )
31	28, 29, 30	sylancr 695	. . . . . . 7 |- ( ( R " y ) e. _V -> ( ( ( x e. _V |-> ( R " x ) ) ` y ) = z <-> y ( x e. _V |-> ( R " x ) ) z ) )
32	25, 31	bitr3d 270	. . . . . 6 |- ( ( R " y ) e. _V -> ( ( R " y ) = z <-> y ( x e. _V |-> ( R " x ) ) z ) )
33	22, 32	syl5bb 272	. . . . 5 |- ( ( R " y ) e. _V -> ( z = ( R " y ) <-> y ( x e. _V |-> ( R " x ) ) z ) )
34	21, 33	syl5bb 272	. . . 4 |- ( ( R " y ) e. _V -> ( yImage R z <-> y ( x e. _V |-> ( R " x ) ) z ) )
35	20, 34	sylbi 207	. . 3 |- ( y e. { x | ( R " x ) e. _V } -> ( yImage R z <-> y ( x e. _V |-> ( R " x ) ) z ) )
36	11, 17, 35	pm5.21nii 368	. 2 |- ( yImage R z <-> y ( x e. _V |-> ( R " x ) ) z )
37	3, 4, 36	eqbrriv 5215	1 |- Image R = ( x e. _V |-> ( R " x ) )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   "cima 5117   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  Imagecimage 31947

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  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-symdif 3844  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-mpt 4730  df-id 5024  df-eprel 5029  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-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-image 31971

This theorem is referenced by:  fvimage  32038