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Theorem funpartfun 30710
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 5132 . 2  |-  Rel  ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
2 vex 3048 . . . . . . 7  |-  z  e. 
_V
32brres 5111 . . . . . 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 462 . . . . 5  |-  ( x ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) z  ->  x F
z )
5 vex 3048 . . . . . . . 8  |-  y  e. 
_V
65brres 5111 . . . . . . 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 452 . . . . . . . 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 30709 . . . . . . . . 9  |-  ( x  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. w
( F " {
x } )  =  { w } )
98anbi1i 701 . . . . . . . 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 253 . . . . . . 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 253 . . . . . 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 4403 . . . . . . . . . . 11  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
13 df-br 4403 . . . . . . . . . . 11  |-  ( x F z  <->  <. x ,  z >.  e.  F
)
1412, 13anbi12i 703 . . . . . . . . . 10  |-  ( ( x F y  /\  x F z )  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) )
15 vex 3048 . . . . . . . . . . . 12  |-  x  e. 
_V
1615, 5elimasn 5193 . . . . . . . . . . 11  |-  ( y  e.  ( F " { x } )  <->  <. x ,  y >.  e.  F )
1715, 2elimasn 5193 . . . . . . . . . . 11  |-  ( z  e.  ( F " { x } )  <->  <. x ,  z >.  e.  F )
1816, 17anbi12i 703 . . . . . . . . . 10  |-  ( ( y  e.  ( F
" { x }
)  /\  z  e.  ( F " { x } ) )  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) )
1914, 18bitr4i 256 . . . . . . . . 9  |-  ( ( x F y  /\  x F z )  <->  ( y  e.  ( F " {
x } )  /\  z  e.  ( F " { x } ) ) )
20 eleq2 2518 . . . . . . . . . . 11  |-  ( ( F " { x } )  =  {
w }  ->  (
y  e.  ( F
" { x }
)  <->  y  e.  {
w } ) )
21 eleq2 2518 . . . . . . . . . . 11  |-  ( ( F " { x } )  =  {
w }  ->  (
z  e.  ( F
" { x }
)  <->  z  e.  {
w } ) )
2220, 21anbi12d 717 . . . . . . . . . 10  |-  ( ( F " { x } )  =  {
w }  ->  (
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) )  <-> 
( y  e.  {
w }  /\  z  e.  { w } ) ) )
23 elsn 3982 . . . . . . . . . . 11  |-  ( y  e.  { w }  <->  y  =  w )
24 elsn 3982 . . . . . . . . . . 11  |-  ( z  e.  { w }  <->  z  =  w )
25 equtr2 1869 . . . . . . . . . . 11  |-  ( ( y  =  w  /\  z  =  w )  ->  y  =  z )
2623, 24, 25syl2anb 482 . . . . . . . . . 10  |-  ( ( y  e.  { w }  /\  z  e.  {
w } )  -> 
y  =  z )
2722, 26syl6bi 232 . . . . . . . . 9  |-  ( ( F " { x } )  =  {
w }  ->  (
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) )  ->  y  =  z ) )
2819, 27syl5bi 221 . . . . . . . 8  |-  ( ( F " { x } )  =  {
w }  ->  (
( x F y  /\  x F z )  ->  y  =  z ) )
2928exlimiv 1776 . . . . . . 7  |-  ( E. w ( F " { x } )  =  { w }  ->  ( ( x F y  /\  x F z )  ->  y  =  z ) )
3029impl 626 . . . . . 6  |-  ( ( ( E. w ( F " { x } )  =  {
w }  /\  x F y )  /\  x F z )  -> 
y  =  z )
3111, 30sylanb 475 . . . . 5  |-  ( ( x ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  /\  x F z )  -> 
y  =  z )
324, 31sylan2 477 . . . 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 1670 . . 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 1669 . 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 30640 . . . 4  |- Funpart F  =  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
3635funeqi 5602 . . 3  |-  ( Fun Funpart F 
<->  Fun  ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) )
37 dffun2 5592 . . 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 253 . 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 931 1  |-  Fun Funpart F
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371   A.wal 1442    = wceq 1444   E.wex 1663    e. wcel 1887   _Vcvv 3045    i^i cin 3403   {csn 3968   <.cop 3974   class class class wbr 4402    X. cxp 4832   dom cdm 4834    |` cres 4836   "cima 4837    o. ccom 4838   Rel wrel 4839   Fun wfun 5576  Singletoncsingle 30604   Singletonscsingles 30605  Imagecimage 30606  Funpartcfunpart 30615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-symdif 3663  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-eprel 4745  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fo 5588  df-fv 5590  df-1st 6793  df-2nd 6794  df-txp 30620  df-singleton 30628  df-singles 30629  df-image 30630  df-funpart 30640
This theorem is referenced by:  fullfunfnv  30713  fullfunfv  30714
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