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Theorem funpartfun 29568
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 5291 . 2  |-  Rel  ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
2 vex 3098 . . . . . . 7  |-  z  e. 
_V
32brres 5270 . . . . . 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 460 . . . . 5  |-  ( x ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) z  ->  x F
z )
5 vex 3098 . . . . . . . 8  |-  y  e. 
_V
65brres 5270 . . . . . . 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 450 . . . . . . . 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 29567 . . . . . . . . 9  |-  ( x  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. w
( F " {
x } )  =  { w } )
98anbi1i 695 . . . . . . . 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 249 . . . . . . 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 249 . . . . . 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 4438 . . . . . . . . . . 11  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
13 df-br 4438 . . . . . . . . . . 11  |-  ( x F z  <->  <. x ,  z >.  e.  F
)
1412, 13anbi12i 697 . . . . . . . . . 10  |-  ( ( x F y  /\  x F z )  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) )
15 vex 3098 . . . . . . . . . . . 12  |-  x  e. 
_V
1615, 5elimasn 5352 . . . . . . . . . . 11  |-  ( y  e.  ( F " { x } )  <->  <. x ,  y >.  e.  F )
1715, 2elimasn 5352 . . . . . . . . . . 11  |-  ( z  e.  ( F " { x } )  <->  <. x ,  z >.  e.  F )
1816, 17anbi12i 697 . . . . . . . . . 10  |-  ( ( y  e.  ( F
" { x }
)  /\  z  e.  ( F " { x } ) )  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) )
1914, 18bitr4i 252 . . . . . . . . 9  |-  ( ( x F y  /\  x F z )  <->  ( y  e.  ( F " {
x } )  /\  z  e.  ( F " { x } ) ) )
20 eleq2 2516 . . . . . . . . . . 11  |-  ( ( F " { x } )  =  {
w }  ->  (
y  e.  ( F
" { x }
)  <->  y  e.  {
w } ) )
21 eleq2 2516 . . . . . . . . . . 11  |-  ( ( F " { x } )  =  {
w }  ->  (
z  e.  ( F
" { x }
)  <->  z  e.  {
w } ) )
2220, 21anbi12d 710 . . . . . . . . . 10  |-  ( ( F " { x } )  =  {
w }  ->  (
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) )  <-> 
( y  e.  {
w }  /\  z  e.  { w } ) ) )
23 elsn 4028 . . . . . . . . . . 11  |-  ( y  e.  { w }  <->  y  =  w )
24 elsn 4028 . . . . . . . . . . 11  |-  ( z  e.  { w }  <->  z  =  w )
25 equtr2 1788 . . . . . . . . . . 11  |-  ( ( y  =  w  /\  z  =  w )  ->  y  =  z )
2623, 24, 25syl2anb 479 . . . . . . . . . 10  |-  ( ( y  e.  { w }  /\  z  e.  {
w } )  -> 
y  =  z )
2722, 26syl6bi 228 . . . . . . . . 9  |-  ( ( F " { x } )  =  {
w }  ->  (
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) )  ->  y  =  z ) )
2819, 27syl5bi 217 . . . . . . . 8  |-  ( ( F " { x } )  =  {
w }  ->  (
( x F y  /\  x F z )  ->  y  =  z ) )
2928exlimiv 1709 . . . . . . 7  |-  ( E. w ( F " { x } )  =  { w }  ->  ( ( x F y  /\  x F z )  ->  y  =  z ) )
3029impl 620 . . . . . 6  |-  ( ( ( E. w ( F " { x } )  =  {
w }  /\  x F y )  /\  x F z )  -> 
y  =  z )
3111, 30sylanb 472 . . . . 5  |-  ( ( x ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  /\  x F z )  -> 
y  =  z )
324, 31sylan2 474 . . . 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 1606 . . 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 1605 . 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 29498 . . . 4  |- Funpart F  =  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
3635funeqi 5598 . . 3  |-  ( Fun Funpart F 
<->  Fun  ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) )
37 dffun2 5588 . . 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 249 . 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 920 1  |-  Fun Funpart F
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1381    = wceq 1383   E.wex 1599    e. wcel 1804   _Vcvv 3095    i^i cin 3460   {csn 4014   <.cop 4020   class class class wbr 4437    X. cxp 4987   dom cdm 4989    |` cres 4991   "cima 4992    o. ccom 4993   Rel wrel 4994   Fun wfun 5572  Singletoncsingle 29462   Singletonscsingles 29463  Imagecimage 29464  Funpartcfunpart 29473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-eprel 4781  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-1st 6785  df-2nd 6786  df-symdif 29443  df-txp 29478  df-singleton 29486  df-singles 29487  df-image 29488  df-funpart 29498
This theorem is referenced by:  fullfunfnv  29571  fullfunfv  29572
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