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Theorem funpartfun 25781
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 5167 . 2  |-  Rel  ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
2 vex 2952 . . . . . . 7  |-  z  e. 
_V
32brres 5145 . . . . . 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 447 . . . . 5  |-  ( x ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) z  ->  x F
z )
5 vex 2952 . . . . . . . 8  |-  y  e. 
_V
65brres 5145 . . . . . . 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 438 . . . . . . . 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 25780 . . . . . . . . 9  |-  ( x  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. w
( F " {
x } )  =  { w } )
98anbi1i 677 . . . . . . . 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 241 . . . . . . 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 241 . . . . . 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 4206 . . . . . . . . . . 11  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
13 df-br 4206 . . . . . . . . . . 11  |-  ( x F z  <->  <. x ,  z >.  e.  F
)
1412, 13anbi12i 679 . . . . . . . . . 10  |-  ( ( x F y  /\  x F z )  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) )
15 vex 2952 . . . . . . . . . . . 12  |-  x  e. 
_V
1615, 5elimasn 5222 . . . . . . . . . . 11  |-  ( y  e.  ( F " { x } )  <->  <. x ,  y >.  e.  F )
1715, 2elimasn 5222 . . . . . . . . . . 11  |-  ( z  e.  ( F " { x } )  <->  <. x ,  z >.  e.  F )
1816, 17anbi12i 679 . . . . . . . . . 10  |-  ( ( y  e.  ( F
" { x }
)  /\  z  e.  ( F " { x } ) )  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) )
1914, 18bitr4i 244 . . . . . . . . 9  |-  ( ( x F y  /\  x F z )  <->  ( y  e.  ( F " {
x } )  /\  z  e.  ( F " { x } ) ) )
20 eleq2 2497 . . . . . . . . . . 11  |-  ( ( F " { x } )  =  {
w }  ->  (
y  e.  ( F
" { x }
)  <->  y  e.  {
w } ) )
21 eleq2 2497 . . . . . . . . . . 11  |-  ( ( F " { x } )  =  {
w }  ->  (
z  e.  ( F
" { x }
)  <->  z  e.  {
w } ) )
2220, 21anbi12d 692 . . . . . . . . . 10  |-  ( ( F " { x } )  =  {
w }  ->  (
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) )  <-> 
( y  e.  {
w }  /\  z  e.  { w } ) ) )
23 elsn 3822 . . . . . . . . . . 11  |-  ( y  e.  { w }  <->  y  =  w )
24 elsn 3822 . . . . . . . . . . 11  |-  ( z  e.  { w }  <->  z  =  w )
25 equtr2 1700 . . . . . . . . . . 11  |-  ( ( y  =  w  /\  z  =  w )  ->  y  =  z )
2623, 24, 25syl2anb 466 . . . . . . . . . 10  |-  ( ( y  e.  { w }  /\  z  e.  {
w } )  -> 
y  =  z )
2722, 26syl6bi 220 . . . . . . . . 9  |-  ( ( F " { x } )  =  {
w }  ->  (
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) )  ->  y  =  z ) )
2819, 27syl5bi 209 . . . . . . . 8  |-  ( ( F " { x } )  =  {
w }  ->  (
( x F y  /\  x F z )  ->  y  =  z ) )
2928exlimiv 1644 . . . . . . 7  |-  ( E. w ( F " { x } )  =  { w }  ->  ( ( x F y  /\  x F z )  ->  y  =  z ) )
3029impl 604 . . . . . 6  |-  ( ( ( E. w ( F " { x } )  =  {
w }  /\  x F y )  /\  x F z )  -> 
y  =  z )
3111, 30sylanb 459 . . . . 5  |-  ( ( x ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  /\  x F z )  -> 
y  =  z )
324, 31sylan2 461 . . . 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 1556 . . 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 1555 . 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 25711 . . . 4  |- Funpart F  =  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
3635funeqi 5467 . . 3  |-  ( Fun Funpart F 
<->  Fun  ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) )
37 dffun2 5457 . . 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 241 . 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 887 1  |-  Fun Funpart F
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2949    i^i cin 3312   {csn 3807   <.cop 3810   class class class wbr 4205    X. cxp 4869   dom cdm 4871    |` cres 4873   "cima 4874    o. ccom 4875   Rel wrel 4876   Fun wfun 5441  Singletoncsingle 25675   Singletonscsingles 25676  Imagecimage 25677  Funpartcfunpart 25686
This theorem is referenced by:  fullfunfnv  25784  fullfunfv  25785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-eprel 4487  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fo 5453  df-fv 5455  df-1st 6342  df-2nd 6343  df-symdif 25656  df-txp 25691  df-singleton 25699  df-singles 25700  df-image 25701  df-funpart 25711
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