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Theorem funpartfun 30781
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 5138 . 2  |-  Rel  ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
2 vex 3034 . . . . . . 7  |-  z  e. 
_V
32brres 5117 . . . . . 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 467 . . . . 5  |-  ( x ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) z  ->  x F
z )
5 vex 3034 . . . . . . . 8  |-  y  e. 
_V
65brres 5117 . . . . . . 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 457 . . . . . . . 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 30780 . . . . . . . . 9  |-  ( x  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. w
( F " {
x } )  =  { w } )
98anbi1i 709 . . . . . . . 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 257 . . . . . . 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 257 . . . . . 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 4396 . . . . . . . . . . 11  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
13 df-br 4396 . . . . . . . . . . 11  |-  ( x F z  <->  <. x ,  z >.  e.  F
)
1412, 13anbi12i 711 . . . . . . . . . 10  |-  ( ( x F y  /\  x F z )  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) )
15 vex 3034 . . . . . . . . . . . 12  |-  x  e. 
_V
1615, 5elimasn 5199 . . . . . . . . . . 11  |-  ( y  e.  ( F " { x } )  <->  <. x ,  y >.  e.  F )
1715, 2elimasn 5199 . . . . . . . . . . 11  |-  ( z  e.  ( F " { x } )  <->  <. x ,  z >.  e.  F )
1816, 17anbi12i 711 . . . . . . . . . 10  |-  ( ( y  e.  ( F
" { x }
)  /\  z  e.  ( F " { x } ) )  <->  ( <. x ,  y >.  e.  F  /\  <. x ,  z
>.  e.  F ) )
1914, 18bitr4i 260 . . . . . . . . 9  |-  ( ( x F y  /\  x F z )  <->  ( y  e.  ( F " {
x } )  /\  z  e.  ( F " { x } ) ) )
20 eleq2 2538 . . . . . . . . . . 11  |-  ( ( F " { x } )  =  {
w }  ->  (
y  e.  ( F
" { x }
)  <->  y  e.  {
w } ) )
21 eleq2 2538 . . . . . . . . . . 11  |-  ( ( F " { x } )  =  {
w }  ->  (
z  e.  ( F
" { x }
)  <->  z  e.  {
w } ) )
2220, 21anbi12d 725 . . . . . . . . . 10  |-  ( ( F " { x } )  =  {
w }  ->  (
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) )  <-> 
( y  e.  {
w }  /\  z  e.  { w } ) ) )
23 elsn 3973 . . . . . . . . . . 11  |-  ( y  e.  { w }  <->  y  =  w )
24 elsn 3973 . . . . . . . . . . 11  |-  ( z  e.  { w }  <->  z  =  w )
25 equtr2 1877 . . . . . . . . . . 11  |-  ( ( y  =  w  /\  z  =  w )  ->  y  =  z )
2623, 24, 25syl2anb 487 . . . . . . . . . 10  |-  ( ( y  e.  { w }  /\  z  e.  {
w } )  -> 
y  =  z )
2722, 26syl6bi 236 . . . . . . . . 9  |-  ( ( F " { x } )  =  {
w }  ->  (
( y  e.  ( F " { x } )  /\  z  e.  ( F " {
x } ) )  ->  y  =  z ) )
2819, 27syl5bi 225 . . . . . . . 8  |-  ( ( F " { x } )  =  {
w }  ->  (
( x F y  /\  x F z )  ->  y  =  z ) )
2928exlimiv 1784 . . . . . . 7  |-  ( E. w ( F " { x } )  =  { w }  ->  ( ( x F y  /\  x F z )  ->  y  =  z ) )
3029impl 632 . . . . . 6  |-  ( ( ( E. w ( F " { x } )  =  {
w }  /\  x F y )  /\  x F z )  -> 
y  =  z )
3111, 30sylanb 480 . . . . 5  |-  ( ( x ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) y  /\  x F z )  -> 
y  =  z )
324, 31sylan2 482 . . . 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 1678 . . 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 1677 . 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 30711 . . . 4  |- Funpart F  =  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
3635funeqi 5609 . . 3  |-  ( Fun Funpart F 
<->  Fun  ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) )
37 dffun2 5599 . . 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 257 . 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 934 1  |-  Fun Funpart F
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031    i^i cin 3389   {csn 3959   <.cop 3965   class class class wbr 4395    X. cxp 4837   dom cdm 4839    |` cres 4841   "cima 4842    o. ccom 4843   Rel wrel 4844   Fun wfun 5583  Singletoncsingle 30675   Singletonscsingles 30676  Imagecimage 30677  Funpartcfunpart 30686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-symdif 3654  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-eprel 4750  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595  df-fv 5597  df-1st 6812  df-2nd 6813  df-txp 30691  df-singleton 30699  df-singles 30700  df-image 30701  df-funpart 30711
This theorem is referenced by:  fullfunfnv  30784  fullfunfv  30785
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