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Theorem funpartfun 24481
Description: The functional part of  F is a function. (Contributed by Scott Fenton, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funpartfun  |-  Fun Funpart F

Proof of Theorem funpartfun
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 4983 . . 3  |-  Rel  ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
2 vex 2791 . . . . . . . 8  |-  z  e. 
_V
32brres 4961 . . . . . . 7  |-  ( x ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) z  <->  ( x F z  /\  x  e. 
dom  ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) ) )
43simplbi 446 . . . . . 6  |-  ( x ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) z  ->  x F
z )
5 vex 2791 . . . . . . . . 9  |-  y  e. 
_V
65brres 4961 . . . . . . . 8  |-  ( x ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  <->  ( x F y  /\  x  e. 
dom  ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) ) )
7 ancom 437 . . . . . . . 8  |-  ( ( x F y  /\  x  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )  <-> 
( x  e.  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  /\  x F y ) )
86, 7bitri 240 . . . . . . 7  |-  ( x ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  <->  ( x  e. 
dom  ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) )  /\  x F y ) )
9 vex 2791 . . . . . . . . . . 11  |-  x  e. 
_V
109eldm 4876 . . . . . . . . . 10  |-  ( x  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. y  x ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) y )
11 brin 4070 . . . . . . . . . . . 12  |-  ( x ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) y  <->  ( x
(Image F  o. Singleton ) y  /\  x ( _V 
X.  Singletons ) y ) )
129, 5brco 4852 . . . . . . . . . . . . . 14  |-  ( x (Image F  o. Singleton ) y  <->  E. z ( xSingleton z  /\  zImage F y ) )
139, 2brsingle 24456 . . . . . . . . . . . . . . . 16  |-  ( xSingleton
z  <->  z  =  {
x } )
142, 5brimage 24465 . . . . . . . . . . . . . . . 16  |-  ( zImage
F y  <->  y  =  ( F " z ) )
1513, 14anbi12i 678 . . . . . . . . . . . . . . 15  |-  ( ( xSingleton z  /\  zImage F y )  <->  ( z  =  { x }  /\  y  =  ( F " z ) ) )
1615exbii 1569 . . . . . . . . . . . . . 14  |-  ( E. z ( xSingleton z  /\  zImage F y )  <->  E. z ( z  =  { x }  /\  y  =  ( F " z ) ) )
17 snex 4216 . . . . . . . . . . . . . . 15  |-  { x }  e.  _V
18 imaeq2 5008 . . . . . . . . . . . . . . . 16  |-  ( z  =  { x }  ->  ( F " z
)  =  ( F
" { x }
) )
1918eqeq2d 2294 . . . . . . . . . . . . . . 15  |-  ( z  =  { x }  ->  ( y  =  ( F " z )  <-> 
y  =  ( F
" { x }
) ) )
2017, 19ceqsexv 2823 . . . . . . . . . . . . . 14  |-  ( E. z ( z  =  { x }  /\  y  =  ( F " z ) )  <->  y  =  ( F " { x } ) )
2112, 16, 203bitri 262 . . . . . . . . . . . . 13  |-  ( x (Image F  o. Singleton ) y  <-> 
y  =  ( F
" { x }
) )
22 brxp 4720 . . . . . . . . . . . . . . 15  |-  ( x ( _V  X.  Singletons ) y  <->  ( x  e.  _V  /\  y  e.  Singletons
) )
239, 22mpbiran 884 . . . . . . . . . . . . . 14  |-  ( x ( _V  X.  Singletons ) y  <->  y  e.  Singletons )
24 elsingles 24457 . . . . . . . . . . . . . 14  |-  ( y  e.  Singletons 
<->  E. w  y  =  { w } )
2523, 24bitri 240 . . . . . . . . . . . . 13  |-  ( x ( _V  X.  Singletons ) y  <->  E. w  y  =  { w } )
2621, 25anbi12i 678 . . . . . . . . . . . 12  |-  ( ( x (Image F  o. Singleton ) y  /\  x ( _V  X.  Singletons ) y )  <-> 
( y  =  ( F " { x } )  /\  E. w  y  =  {
w } ) )
27 ancom 437 . . . . . . . . . . . . 13  |-  ( ( y  =  ( F
" { x }
)  /\  E. w  y  =  { w } )  <->  ( E. w  y  =  {
w }  /\  y  =  ( F " { x } ) ) )
28 19.41v 1842 . . . . . . . . . . . . 13  |-  ( E. w ( y  =  { w }  /\  y  =  ( F " { x } ) )  <->  ( E. w  y  =  { w }  /\  y  =  ( F " { x } ) ) )
2927, 28bitr4i 243 . . . . . . . . . . . 12  |-  ( ( y  =  ( F
" { x }
)  /\  E. w  y  =  { w } )  <->  E. w
( y  =  {
w }  /\  y  =  ( F " { x } ) ) )
3011, 26, 293bitri 262 . . . . . . . . . . 11  |-  ( x ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) y  <->  E. w
( y  =  {
w }  /\  y  =  ( F " { x } ) ) )
3130exbii 1569 . . . . . . . . . 10  |-  ( E. y  x ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) y  <->  E. y E. w ( y  =  { w }  /\  y  =  ( F " { x } ) ) )
32 excom 1786 . . . . . . . . . . 11  |-  ( E. y E. w ( y  =  { w }  /\  y  =  ( F " { x } ) )  <->  E. w E. y ( y  =  { w }  /\  y  =  ( F " { x } ) ) )
33 snex 4216 . . . . . . . . . . . . 13  |-  { w }  e.  _V
34 eqeq1 2289 . . . . . . . . . . . . 13  |-  ( y  =  { w }  ->  ( y  =  ( F " { x } )  <->  { w }  =  ( F " { x } ) ) )
3533, 34ceqsexv 2823 . . . . . . . . . . . 12  |-  ( E. y ( y  =  { w }  /\  y  =  ( F " { x } ) )  <->  { w }  =  ( F " { x } ) )
3635exbii 1569 . . . . . . . . . . 11  |-  ( E. w E. y ( y  =  { w }  /\  y  =  ( F " { x } ) )  <->  E. w { w }  =  ( F " { x } ) )
3732, 36bitri 240 . . . . . . . . . 10  |-  ( E. y E. w ( y  =  { w }  /\  y  =  ( F " { x } ) )  <->  E. w { w }  =  ( F " { x } ) )
3810, 31, 373bitri 262 . . . . . . . . 9  |-  ( x  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. w { w }  =  ( F " { x } ) )
399, 5elimasn 5038 . . . . . . . . . . . . . 14  |-  ( y  e.  ( F " { x } )  <->  <. x ,  y >.  e.  F )
40 df-br 4024 . . . . . . . . . . . . . 14  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
4139, 40bitr4i 243 . . . . . . . . . . . . 13  |-  ( y  e.  ( F " { x } )  <-> 
x F y )
429, 2elimasn 5038 . . . . . . . . . . . . . 14  |-  ( z  e.  ( F " { x } )  <->  <. x ,  z >.  e.  F )
43 df-br 4024 . . . . . . . . . . . . . 14  |-  ( x F z  <->  <. x ,  z >.  e.  F
)
4442, 43bitr4i 243 . . . . . . . . . . . . 13  |-  ( z  e.  ( F " { x } )  <-> 
x F z )
4541, 44anbi12i 678 . . . . . . . . . . . 12  |-  ( ( y  e.  ( F
" { x }
)  /\  z  e.  ( F " { x } ) )  <->  ( x F y  /\  x F z ) )
46 eleq2 2344 . . . . . . . . . . . . . 14  |-  ( { w }  =  ( F " { x } )  ->  (
y  e.  { w } 
<->  y  e.  ( F
" { x }
) ) )
47 eleq2 2344 . . . . . . . . . . . . . 14  |-  ( { w }  =  ( F " { x } )  ->  (
z  e.  { w } 
<->  z  e.  ( F
" { x }
) ) )
4846, 47anbi12d 691 . . . . . . . . . . . . 13  |-  ( { w }  =  ( F " { x } )  ->  (
( y  e.  {
w }  /\  z  e.  { w } )  <-> 
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) ) ) )
49 elsn 3655 . . . . . . . . . . . . . 14  |-  ( y  e.  { w }  <->  y  =  w )
50 elsn 3655 . . . . . . . . . . . . . 14  |-  ( z  e.  { w }  <->  z  =  w )
51 equtr2 1654 . . . . . . . . . . . . . 14  |-  ( ( y  =  w  /\  z  =  w )  ->  y  =  z )
5249, 50, 51syl2anb 465 . . . . . . . . . . . . 13  |-  ( ( y  e.  { w }  /\  z  e.  {
w } )  -> 
y  =  z )
5348, 52syl6bir 220 . . . . . . . . . . . 12  |-  ( { w }  =  ( F " { x } )  ->  (
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) )  ->  y  =  z ) )
5445, 53syl5bir 209 . . . . . . . . . . 11  |-  ( { w }  =  ( F " { x } )  ->  (
( x F y  /\  x F z )  ->  y  =  z ) )
5554exlimiv 1666 . . . . . . . . . 10  |-  ( E. w { w }  =  ( F " { x } )  ->  ( ( x F y  /\  x F z )  -> 
y  =  z ) )
5655imp 418 . . . . . . . . 9  |-  ( ( E. w { w }  =  ( F " { x } )  /\  ( x F y  /\  x F z ) )  -> 
y  =  z )
5738, 56sylanb 458 . . . . . . . 8  |-  ( ( x  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  /\  ( x F y  /\  x F z ) )  ->  y  =  z )
5857anassrs 629 . . . . . . 7  |-  ( ( ( x  e.  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  /\  x F y )  /\  x F z )  -> 
y  =  z )
598, 58sylanb 458 . . . . . 6  |-  ( ( x ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  /\  x F z )  -> 
y  =  z )
604, 59sylan2 460 . . . . 5  |-  ( ( x ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  /\  x ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) z )  ->  y  =  z )
6160ax-gen 1533 . . . 4  |-  A. z
( ( x ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  /\  x ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) z )  ->  y  =  z )
6261gen2 1534 . . 3  |-  A. x A. y A. z ( ( x ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) ) y  /\  x ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) z )  ->  y  =  z )
63 dffun2 5265 . . 3  |-  ( Fun  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )  <-> 
( Rel  ( F  |` 
dom  ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) )  /\  A. x A. y A. z ( ( x ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  /\  x ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) z )  ->  y  =  z ) ) )
641, 62, 63mpbir2an 886 . 2  |-  Fun  ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
65 df-funpart 24415 . . 3  |- Funpart F  =  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
6665funeqi 5275 . 2  |-  ( Fun Funpart F 
<->  Fun  ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) )
6764, 66mpbir 200 1  |-  Fun Funpart F
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   {csn 3640   <.cop 3643   class class class wbr 4023    X. cxp 4687   dom cdm 4689    |` cres 4691   "cima 4692    o. ccom 4693   Rel wrel 4694   Fun wfun 5249  Singletoncsingle 24381   Singletonscsingles 24382  Imagecimage 24383  Funpartcfunpart 24392
This theorem is referenced by:  fullfunfnv  24484  fullfunfv  24485
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-singleton 24403  df-singles 24404  df-image 24405  df-funpart 24415
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