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Theorem funimage 30239
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 3569 . . . 4  |-  ( ( _V  X.  _V )  \  ran  ( ( _V 
(x)  _E  )  /_\  ( (  _E  o.  `' A
)  (x)  _V )
) )  C_  ( _V  X.  _V )
2 df-rel 4949 . . . 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 30174 . . . 4  |- Image A  =  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )  /_\  ( (  _E  o.  `' A ) 
(x)  _V ) ) )
54releqi 5028 . . 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 3061 . . . . . 6  |-  x  e. 
_V
8 vex 3061 . . . . . 6  |-  y  e. 
_V
97, 8brimage 30237 . . . . 5  |-  ( xImage
A y  <->  y  =  ( A " x ) )
10 vex 3061 . . . . . 6  |-  z  e. 
_V
117, 10brimage 30237 . . . . 5  |-  ( xImage
A z  <->  z  =  ( A " x ) )
12 eqtr3 2430 . . . . 5  |-  ( ( y  =  ( A
" x )  /\  z  =  ( A " x ) )  -> 
y  =  z )
139, 11, 12syl2anb 477 . . . 4  |-  ( ( xImage A y  /\  xImage A z )  -> 
y  =  z )
1413gen2 1640 . . 3  |-  A. y A. z ( ( xImage
A y  /\  xImage A z )  ->  y  =  z )
1514ax-gen 1639 . 2  |-  A. x A. y A. z ( ( xImage A y  /\  xImage A z )  ->  y  =  z )
16 dffun2 5535 . 2  |-  ( Fun Image A 
<->  ( Rel Image A  /\  A. x A. y A. z
( ( xImage A
y  /\  xImage A
z )  ->  y  =  z ) ) )
176, 15, 16mpbir2an 921 1  |-  Fun Image A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1403    = wceq 1405   _Vcvv 3058    \ cdif 3410    C_ wss 3413    /_\ csymdif 3668   class class class wbr 4394    _E cep 4731    X. cxp 4940   `'ccnv 4941   ran crn 4943   "cima 4945    o. ccom 4946   Rel wrel 4947   Fun wfun 5519    (x) ctxp 30140  Imagecimage 30150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-symdif 3669  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-eprel 4733  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fo 5531  df-fv 5533  df-1st 6738  df-2nd 6739  df-txp 30164  df-image 30174
This theorem is referenced by:  fnimage  30240  imageval  30241  imagesset  30264
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