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Theorem funimage 28126
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 3594 . . . 4  |-  ( ( _V  X.  _V )  \  ran  ( ( _V 
(x)  _E  )(++) (
(  _E  o.  `' A )  (x)  _V ) ) )  C_  ( _V  X.  _V )
2 df-rel 4958 . . . 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 209 . . 3  |-  Rel  (
( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) ( (  _E  o.  `' A )  (x)  _V ) ) )
4 df-image 28061 . . . 4  |- Image A  =  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) ( (  _E  o.  `' A )  (x)  _V ) ) )
54releqi 5034 . . 3  |-  ( Rel Image A 
<->  Rel  ( ( _V 
X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) ( (  _E  o.  `' A ) 
(x)  _V ) ) ) )
63, 5mpbir 209 . 2  |-  Rel Image A
7 vex 3081 . . . . . 6  |-  x  e. 
_V
8 vex 3081 . . . . . 6  |-  y  e. 
_V
97, 8brimage 28124 . . . . 5  |-  ( xImage
A y  <->  y  =  ( A " x ) )
10 vex 3081 . . . . . 6  |-  z  e. 
_V
117, 10brimage 28124 . . . . 5  |-  ( xImage
A z  <->  z  =  ( A " x ) )
12 eqtr3 2482 . . . . 5  |-  ( ( y  =  ( A
" x )  /\  z  =  ( A " x ) )  -> 
y  =  z )
139, 11, 12syl2anb 479 . . . 4  |-  ( ( xImage A y  /\  xImage A z )  -> 
y  =  z )
1413gen2 1593 . . 3  |-  A. y A. z ( ( xImage
A y  /\  xImage A z )  ->  y  =  z )
1514ax-gen 1592 . 2  |-  A. x A. y A. z ( ( xImage A y  /\  xImage A z )  ->  y  =  z )
16 dffun2 5539 . 2  |-  ( Fun Image A 
<->  ( Rel Image A  /\  A. x A. y A. z
( ( xImage A
y  /\  xImage A
z )  ->  y  =  z ) ) )
176, 15, 16mpbir2an 911 1  |-  Fun Image A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1368    = wceq 1370   _Vcvv 3078    \ cdif 3436    C_ wss 3439   class class class wbr 4403    _E cep 4741    X. cxp 4949   `'ccnv 4950   ran crn 4952   "cima 4954    o. ccom 4955   Rel wrel 4956   Fun wfun 5523  (++)csymdif 28015    (x) ctxp 28027  Imagecimage 28037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-eprel 4743  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-1st 6690  df-2nd 6691  df-symdif 28016  df-txp 28051  df-image 28061
This theorem is referenced by:  fnimage  28127  imageval  28128  imagesset  28151
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