Theorem funimage 32035

Description: Image A is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
funimage	|- Fun Image A

Proof of Theorem funimage

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

Step	Hyp	Ref	Expression
1		difss 3737	. . . 4 |- ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) ) C_ ( _V X. _V )
2		df-rel 5121	. . . 4 |- ( Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) ) <-> ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) ) C_ ( _V X. _V ) )
3	1, 2	mpbir 221	. . 3 |- Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) )
4		df-image 31971	. . . 4 |- Image A = ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) )
5	4	releqi 5202	. . 3 |- ( Rel Image A <-> Rel ( ( _V X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( _E o. `' A ) (x) _V ) ) ) )
6	3, 5	mpbir 221	. 2 |- Rel Image A
7		vex 3203	. . . . . 6 |- x e. _V
8		vex 3203	. . . . . 6 |- y e. _V
9	7, 8	brimage 32033	. . . . 5 |- ( xImage A y <-> y = ( A " x ) )
10		vex 3203	. . . . . 6 |- z e. _V
11	7, 10	brimage 32033	. . . . 5 |- ( xImage A z <-> z = ( A " x ) )
12		eqtr3 2643	. . . . 5 |- ( ( y = ( A " x ) /\ z = ( A " x ) ) -> y = z )
13	9, 11, 12	syl2anb 496	. . . 4 |- ( ( xImage A y /\ xImage A z ) -> y = z )
14	13	gen2 1723	. . 3 |- A. y A. z ( ( xImage A y /\ xImage A z ) -> y = z )
15	14	ax-gen 1722	. 2 |- A. x A. y A. z ( ( xImage A y /\ xImage A z ) -> y = z )
16		dffun2 5898	. 2 |- ( Fun Image A <-> ( Rel Image A /\ A. x A. y A. z ( ( xImage A y /\ xImage A z ) -> y = z ) ) )
17	6, 15, 16	mpbir2an 955	1 |- Fun Image A

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   _Vcvv 3200   \ cdif 3571   C_ wss 3574   /_\ csymdif 3843   class class class wbr 4653   _E cep 5028   X. cxp 5112   `'ccnv 5113   ran crn 5115   "cima 5117   o. ccom 5118   Rel wrel 5119   Fun wfun 5882   (x) ctxp 31937  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:  fnimage  32036  imageval  32037  imagesset  32060