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Theorem funpartfun 24867
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 5020 . 2  |-  Rel  ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
2 vex 2825 . . . . . . 7  |-  z  e. 
_V
32brres 4998 . . . . . 6  |-  ( 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 . . . . 5  |-  ( x ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) z  ->  x F
z )
5 vex 2825 . . . . . . . 8  |-  y  e. 
_V
65brres 4998 . . . . . . 7  |-  ( 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 ) )
8 funpartlem 24866 . . . . . . . . 9  |-  ( x  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. w
( F " {
x } )  =  { w } )
98anbi1i 676 . . . . . . . 8  |-  ( ( x  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  /\  x F y )  <->  ( E. w ( F " { x } )  =  { w }  /\  x F y ) )
107, 9bitri 240 . . . . . . 7  |-  ( ( x F y  /\  x  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )  <-> 
( E. w ( F " { x } )  =  {
w }  /\  x F y ) )
116, 10bitri 240 . . . . . 6  |-  ( x ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  <->  ( E. w
( F " {
x } )  =  { w }  /\  x F y ) )
12 df-br 4061 . . . . . . . . . . 11  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
13 df-br 4061 . . . . . . . . . . 11  |-  ( x F z  <->  <. x ,  z >.  e.  F
)
1412, 13anbi12i 678 . . . . . . . . . 10  |-  ( ( x F y  /\  x F z )  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) )
15 vex 2825 . . . . . . . . . . . 12  |-  x  e. 
_V
1615, 5elimasn 5075 . . . . . . . . . . 11  |-  ( y  e.  ( F " { x } )  <->  <. x ,  y >.  e.  F )
1715, 2elimasn 5075 . . . . . . . . . . 11  |-  ( z  e.  ( F " { x } )  <->  <. x ,  z >.  e.  F )
1816, 17anbi12i 678 . . . . . . . . . 10  |-  ( ( y  e.  ( F
" { x }
)  /\  z  e.  ( F " { x } ) )  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) )
1914, 18bitr4i 243 . . . . . . . . 9  |-  ( ( x F y  /\  x F z )  <->  ( y  e.  ( F " {
x } )  /\  z  e.  ( F " { x } ) ) )
20 eleq2 2377 . . . . . . . . . . 11  |-  ( ( F " { x } )  =  {
w }  ->  (
y  e.  ( F
" { x }
)  <->  y  e.  {
w } ) )
21 eleq2 2377 . . . . . . . . . . 11  |-  ( ( F " { x } )  =  {
w }  ->  (
z  e.  ( F
" { x }
)  <->  z  e.  {
w } ) )
2220, 21anbi12d 691 . . . . . . . . . 10  |-  ( ( F " { x } )  =  {
w }  ->  (
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) )  <-> 
( y  e.  {
w }  /\  z  e.  { w } ) ) )
23 elsn 3689 . . . . . . . . . . 11  |-  ( y  e.  { w }  <->  y  =  w )
24 elsn 3689 . . . . . . . . . . 11  |-  ( z  e.  { w }  <->  z  =  w )
25 equtr2 1679 . . . . . . . . . . 11  |-  ( ( y  =  w  /\  z  =  w )  ->  y  =  z )
2623, 24, 25syl2anb 465 . . . . . . . . . 10  |-  ( ( y  e.  { w }  /\  z  e.  {
w } )  -> 
y  =  z )
2722, 26syl6bi 219 . . . . . . . . 9  |-  ( ( F " { x } )  =  {
w }  ->  (
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) )  ->  y  =  z ) )
2819, 27syl5bi 208 . . . . . . . 8  |-  ( ( F " { x } )  =  {
w }  ->  (
( x F y  /\  x F z )  ->  y  =  z ) )
2928exlimiv 1625 . . . . . . 7  |-  ( E. w ( F " { x } )  =  { w }  ->  ( ( x F y  /\  x F z )  ->  y  =  z ) )
3029impl 603 . . . . . 6  |-  ( ( ( E. w ( F " { x } )  =  {
w }  /\  x F y )  /\  x F z )  -> 
y  =  z )
3111, 30sylanb 458 . . . . 5  |-  ( ( x ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  /\  x F z )  -> 
y  =  z )
324, 31sylan2 460 . . . 4  |-  ( ( 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 )
3332gen2 1538 . . 3  |-  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 )
3433ax-gen 1537 . 2  |-  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 )
35 df-funpart 24800 . . . 4  |- Funpart F  =  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
3635funeqi 5312 . . 3  |-  ( Fun Funpart F 
<->  Fun  ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) )
37 dffun2 5302 . . 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 ) ) )
3836, 37bitri 240 . 2  |-  ( Fun Funpart F 
<->  ( 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 ) ) )
391, 34, 38mpbir2an 886 1  |-  Fun Funpart F
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1531   E.wex 1532    = wceq 1633    e. wcel 1701   _Vcvv 2822    i^i cin 3185   {csn 3674   <.cop 3677   class class class wbr 4060    X. cxp 4724   dom cdm 4726    |` cres 4728   "cima 4729    o. ccom 4730   Rel wrel 4731   Fun wfun 5286  Singletoncsingle 24766   Singletonscsingles 24767  Imagecimage 24768  Funpartcfunpart 24777
This theorem is referenced by:  fullfunfnv  24870  fullfunfv  24871
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-singleton 24788  df-singles 24789  df-image 24790  df-funpart 24800
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