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Theorem funimage 24852
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.
StepHypRef Expression
1 difss 3337 . . . 4  |-  ( ( _V  X.  _V )  \  ran  ( ( _V 
(x)  _E  )(++) (
(  _E  o.  `' A )  (x)  _V ) ) )  C_  ( _V  X.  _V )
2 df-rel 4733 . . . 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 ) )
31, 2mpbir 200 . . 3  |-  Rel  (
( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) ( (  _E  o.  `' A )  (x)  _V ) ) )
4 df-image 24790 . . . 4  |- Image A  =  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) ( (  _E  o.  `' A )  (x)  _V ) ) )
54releqi 4809 . . 3  |-  ( Rel Image A 
<->  Rel  ( ( _V 
X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) ( (  _E  o.  `' A ) 
(x)  _V ) ) ) )
63, 5mpbir 200 . 2  |-  Rel Image A
7 vex 2825 . . . . . 6  |-  x  e. 
_V
8 vex 2825 . . . . . 6  |-  y  e. 
_V
97, 8brimage 24850 . . . . 5  |-  ( xImage
A y  <->  y  =  ( A " x ) )
10 vex 2825 . . . . . 6  |-  z  e. 
_V
117, 10brimage 24850 . . . . 5  |-  ( xImage
A z  <->  z  =  ( A " x ) )
12 eqtr3 2335 . . . . 5  |-  ( ( y  =  ( A
" x )  /\  z  =  ( A " x ) )  -> 
y  =  z )
139, 11, 12syl2anb 465 . . . 4  |-  ( ( xImage A y  /\  xImage A z )  -> 
y  =  z )
1413gen2 1538 . . 3  |-  A. y A. z ( ( xImage
A y  /\  xImage A z )  ->  y  =  z )
1514ax-gen 1537 . 2  |-  A. x A. y A. z ( ( xImage A y  /\  xImage A z )  ->  y  =  z )
16 dffun2 5302 . 2  |-  ( Fun Image A 
<->  ( Rel Image A  /\  A. x A. y A. z
( ( xImage A
y  /\  xImage A
z )  ->  y  =  z ) ) )
176, 15, 16mpbir2an 886 1  |-  Fun Image A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1531    = wceq 1633   _Vcvv 2822    \ cdif 3183    C_ wss 3186   class class class wbr 4060    _E cep 4340    X. cxp 4724   `'ccnv 4725   ran crn 4727   "cima 4729    o. ccom 4730   Rel wrel 4731   Fun wfun 5286  (++)csymdif 24746    (x) ctxp 24758  Imagecimage 24768
This theorem is referenced by:  fnimage  24853  imageval  24854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-eprel 4342  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fo 5298  df-fv 5300  df-1st 6164  df-2nd 6165  df-symdif 24747  df-txp 24780  df-image 24790
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