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Theorem funimage 24467
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 3303 . . . 4  |-  ( ( _V  X.  _V )  \  ran  ( ( _V 
(x)  _E  )(++) (
(  _E  o.  `' A )  (x)  _V ) ) )  C_  ( _V  X.  _V )
2 df-rel 4696 . . . 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 24405 . . . 4  |- Image A  =  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) ( (  _E  o.  `' A )  (x)  _V ) ) )
54releqi 4772 . . 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 2791 . . . . . 6  |-  x  e. 
_V
8 vex 2791 . . . . . 6  |-  y  e. 
_V
97, 8brimage 24465 . . . . 5  |-  ( xImage
A y  <->  y  =  ( A " x ) )
10 vex 2791 . . . . . 6  |-  z  e. 
_V
117, 10brimage 24465 . . . . 5  |-  ( xImage
A z  <->  z  =  ( A " x ) )
12 eqtr3 2302 . . . . 5  |-  ( ( y  =  ( A
" x )  /\  z  =  ( A " x ) )  -> 
y  =  z )
139, 11, 12syl2anb 465 . . . 4  |-  ( ( xImage A y  /\  xImage A z )  -> 
y  =  z )
1413gen2 1534 . . 3  |-  A. y A. z ( ( xImage
A y  /\  xImage A z )  ->  y  =  z )
1514ax-gen 1533 . 2  |-  A. x A. y A. z ( ( xImage A y  /\  xImage A z )  ->  y  =  z )
16 dffun2 5265 . 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 1527    = wceq 1623   _Vcvv 2788    \ cdif 3149    C_ wss 3152   class class class wbr 4023    _E cep 4303    X. cxp 4687   `'ccnv 4688   ran crn 4690   "cima 4692    o. ccom 4693   Rel wrel 4694   Fun wfun 5249  (++)csymdif 24361    (x) ctxp 24373  Imagecimage 24383
This theorem is referenced by:  fnimage  24468  imageval  24469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-image 24405
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