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Theorem funpartfv 24483
Description: The function value of the functional part is identical to the original functional value. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funpartfv  |-  (Funpart F `  A )  =  ( F `  A )

Proof of Theorem funpartfv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-funpart 24415 . . 3  |- Funpart F  =  ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )
21fveq1i 5526 . 2  |-  (Funpart F `  A )  =  ( ( F  |`  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `
 A )
3 fvres 5542 . . 3  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `  A
)  =  ( F `
 A ) )
4 nfvres 5557 . . . 4  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `  A
)  =  (/) )
5 dffv5 24463 . . . . 5  |-  ( F `
 A )  = 
U. U. ( { ( F " { A } ) }  i^i  Singletons )
6 elsn 3655 . . . . . . . . . . 11  |-  ( x  e.  { ( F
" { A }
) }  <->  x  =  ( F " { A } ) )
7 noel 3459 . . . . . . . . . . . . . . . . . . . . 21  |-  -.  x  e.  (/)
8 vex 2791 . . . . . . . . . . . . . . . . . . . . . . 23  |-  x  e. 
_V
98snid 3667 . . . . . . . . . . . . . . . . . . . . . 22  |-  x  e. 
{ x }
10 eleq2 2344 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (/)  =  { x }  ->  ( x  e.  (/)  <->  x  e.  { x } ) )
119, 10mpbiri 224 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (/)  =  { x }  ->  x  e.  (/) )
127, 11mto 167 . . . . . . . . . . . . . . . . . . . 20  |-  -.  (/)  =  {
x }
1312nex 1542 . . . . . . . . . . . . . . . . . . 19  |-  -.  E. x (/)  =  { x }
14 elsingles 24457 . . . . . . . . . . . . . . . . . . 19  |-  ( (/)  e. 
Singletons  <->  E. x (/)  =  {
x } )
1513, 14mtbir 290 . . . . . . . . . . . . . . . . . 18  |-  -.  (/)  e.  Singletons
16 eleq1 2343 . . . . . . . . . . . . . . . . . 18  |-  ( ( F " { A } )  =  (/)  ->  ( ( F " { A } )  e.  Singletons  <->  (/)  e. 
Singletons ) )
1715, 16mtbiri 294 . . . . . . . . . . . . . . . . 17  |-  ( ( F " { A } )  =  (/)  ->  -.  ( F " { A } )  e.  Singletons
)
1817necon2ai 2491 . . . . . . . . . . . . . . . 16  |-  ( ( F " { A } )  e.  Singletons  ->  ( F
" { A }
)  =/=  (/) )
19 imaeq2 5008 . . . . . . . . . . . . . . . . . 18  |-  ( { A }  =  (/)  ->  ( F " { A } )  =  ( F " (/) ) )
20 ima0 5030 . . . . . . . . . . . . . . . . . 18  |-  ( F
" (/) )  =  (/)
2119, 20syl6eq 2331 . . . . . . . . . . . . . . . . 17  |-  ( { A }  =  (/)  ->  ( F " { A } )  =  (/) )
2221necon3i 2485 . . . . . . . . . . . . . . . 16  |-  ( ( F " { A } )  =/=  (/)  ->  { A }  =/=  (/) )
2318, 22syl 15 . . . . . . . . . . . . . . 15  |-  ( ( F " { A } )  e.  Singletons  ->  { A }  =/=  (/) )
24 snprc 3695 . . . . . . . . . . . . . . . . 17  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2524bicomi 193 . . . . . . . . . . . . . . . 16  |-  ( { A }  =  (/)  <->  -.  A  e.  _V )
2625necon2abii 2501 . . . . . . . . . . . . . . 15  |-  ( A  e.  _V  <->  { A }  =/=  (/) )
2723, 26sylibr 203 . . . . . . . . . . . . . 14  |-  ( ( F " { A } )  e.  Singletons  ->  A  e. 
_V )
28 sneq 3651 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  A  ->  { x }  =  { A } )
2928imaeq2d 5012 . . . . . . . . . . . . . . . . 17  |-  ( x  =  A  ->  ( F " { x }
)  =  ( F
" { A }
) )
3029eleq1d 2349 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  (
( F " {
x } )  e.  Singletons  <->  ( F " { A } )  e.  Singletons ) )
31 eleq1 2343 . . . . . . . . . . . . . . . 16  |-  ( x  =  A  ->  (
x  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  A  e.  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) )
3230, 31imbi12d 311 . . . . . . . . . . . . . . 15  |-  ( x  =  A  ->  (
( ( F " { x } )  e.  Singletons  ->  x  e.  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) )  <-> 
( ( F " { A } )  e.  Singletons  ->  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) ) )
33 eqid 2283 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( F
" { x }
)  =  ( F
" { x }
)
34 snex 4216 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  { x }  e.  _V
35 brimageg 24466 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( { x }  e.  _V  /\  ( F " { x } )  e.  Singletons )  ->  ( { x }Image F
( F " {
x } )  <->  ( F " { x } )  =  ( F " { x } ) ) )
3634, 35mpan 651 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( F " { x } )  e.  Singletons  ->  ( { x }Image F ( F " { x } )  <->  ( F " { x } )  =  ( F " { x } ) ) )
3733, 36mpbiri 224 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( F " { x } )  e.  Singletons  ->  { x }Image F ( F " { x } ) )
38 snex 4216 . . . . . . . . . . . . . . . . . . . . . . 23  |-  { x }  e.  _V
39 breq1 4026 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  =  { x }  ->  ( yImage F ( F " { x } )  <->  { x }Image F ( F " { x } ) ) )
4038, 39ceqsexv 2823 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. y ( y  =  { x }  /\  yImage F ( F " { x } ) )  <->  { x }Image F
( F " {
x } ) )
4137, 40sylibr 203 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F " { x } )  e.  Singletons  ->  E. y
( y  =  {
x }  /\  yImage F ( F " { x } ) ) )
42 vex 2791 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  y  e. 
_V
438, 42brsingle 24456 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( xSingleton
y  <->  y  =  {
x } )
44 df-br 4024 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( xSingleton
y  <->  <. x ,  y
>.  e. Singleton )
4543, 44bitr3i 242 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  =  { x }  <->  <.
x ,  y >.  e. Singleton )
46 df-br 4024 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( yImage
F ( F " { x } )  <->  <. y ,  ( F
" { x }
) >.  e. Image F )
4745, 46anbi12i 678 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  =  { x }  /\  yImage F ( F " { x } ) )  <->  ( <. x ,  y >.  e. Singleton  /\  <. y ,  ( F " { x } )
>.  e. Image F ) )
4847exbii 1569 . . . . . . . . . . . . . . . . . . . . 21  |-  ( E. y ( y  =  { x }  /\  yImage F ( F " { x } ) )  <->  E. y ( <.
x ,  y >.  e. Singleton 
/\  <. y ,  ( F " { x } ) >.  e. Image F
) )
4941, 48sylib 188 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F " { x } )  e.  Singletons  ->  E. y
( <. x ,  y
>.  e. Singleton  /\  <. y ,  ( F " { x } ) >.  e. Image F
) )
50 opelco2g 4851 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  _V  /\  ( F " { x } )  e.  Singletons )  ->  ( <. x ,  ( F
" { x }
) >.  e.  (Image F  o. Singleton )  <->  E. y ( <.
x ,  y >.  e. Singleton 
/\  <. y ,  ( F " { x } ) >.  e. Image F
) ) )
518, 50mpan 651 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F " { x } )  e.  Singletons  ->  ( <.
x ,  ( F
" { x }
) >.  e.  (Image F  o. Singleton )  <->  E. y ( <.
x ,  y >.  e. Singleton 
/\  <. y ,  ( F " { x } ) >.  e. Image F
) ) )
5249, 51mpbird 223 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F " { x } )  e.  Singletons  ->  <. x ,  ( F " { x } )
>.  e.  (Image F  o. Singleton ) )
53 df-br 4024 . . . . . . . . . . . . . . . . . . 19  |-  ( x (Image F  o. Singleton ) ( F " { x } )  <->  <. x ,  ( F " {
x } ) >.  e.  (Image F  o. Singleton ) )
5452, 53sylibr 203 . . . . . . . . . . . . . . . . . 18  |-  ( ( F " { x } )  e.  Singletons  ->  x (Image
F  o. Singleton ) ( F
" { x }
) )
55 opelxpi 4721 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  _V  /\  ( F " { x } )  e.  Singletons )  ->  <. x ,  ( F " { x } )
>.  e.  ( _V  X.  Singletons ) )
568, 55mpan 651 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F " { x } )  e.  Singletons  ->  <. x ,  ( F " { x } )
>.  e.  ( _V  X.  Singletons ) )
57 df-br 4024 . . . . . . . . . . . . . . . . . . 19  |-  ( x ( _V  X.  Singletons ) ( F
" { x }
)  <->  <. x ,  ( F " { x } ) >.  e.  ( _V  X.  Singletons ) )
5856, 57sylibr 203 . . . . . . . . . . . . . . . . . 18  |-  ( ( F " { x } )  e.  Singletons  ->  x ( _V  X.  Singletons ) ( F
" { x }
) )
59 brin 4070 . . . . . . . . . . . . . . . . . 18  |-  ( x ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ( F " { x } )  <-> 
( x (Image F  o. Singleton ) ( F " { x } )  /\  x ( _V 
X.  Singletons ) ( F
" { x }
) ) )
6054, 58, 59sylanbrc 645 . . . . . . . . . . . . . . . . 17  |-  ( ( F " { x } )  e.  Singletons  ->  x ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ( F " { x } ) )
61 breq2 4027 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  ( F " { x } )  ->  ( x ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) y  <-> 
x ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) ( F " { x } ) ) )
6261spcegv 2869 . . . . . . . . . . . . . . . . 17  |-  ( ( F " { x } )  e.  Singletons  ->  ( x ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ( F " { x } )  ->  E. y  x ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) y ) )
6360, 62mpd 14 . . . . . . . . . . . . . . . 16  |-  ( ( F " { x } )  e.  Singletons  ->  E. y  x ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) y )
648eldm 4876 . . . . . . . . . . . . . . . 16  |-  ( x  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. y  x ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) y )
6563, 64sylibr 203 . . . . . . . . . . . . . . 15  |-  ( ( F " { x } )  e.  Singletons  ->  x  e. 
dom  ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) )
6632, 65vtoclg 2843 . . . . . . . . . . . . . 14  |-  ( A  e.  _V  ->  (
( F " { A } )  e.  Singletons  ->  A  e. 
dom  ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) ) )
6727, 66mpcom 32 . . . . . . . . . . . . 13  |-  ( ( F " { A } )  e.  Singletons  ->  A  e. 
dom  ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) )
6867con3i 127 . . . . . . . . . . . 12  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  ->  -.  ( F " { A } )  e.  Singletons )
69 eleq1 2343 . . . . . . . . . . . . 13  |-  ( x  =  ( F " { A } )  -> 
( x  e.  Singletons  <->  ( F " { A } )  e.  Singletons
) )
7069notbid 285 . . . . . . . . . . . 12  |-  ( x  =  ( F " { A } )  -> 
( -.  x  e.  Singletons  <->  -.  ( F " { A } )  e.  Singletons ) )
7168, 70syl5ibrcom 213 . . . . . . . . . . 11  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( x  =  ( F " { A } )  ->  -.  x  e.  Singletons ) )
726, 71syl5bi 208 . . . . . . . . . 10  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( x  e.  {
( F " { A } ) }  ->  -.  x  e.  Singletons ) )
7372ralrimiv 2625 . . . . . . . . 9  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  ->  A. x  e.  { ( F " { A } ) }  -.  x  e.  Singletons )
74 disj 3495 . . . . . . . . 9  |-  ( ( { ( F " { A } ) }  i^i  Singletons )  =  (/)  <->  A. x  e.  { ( F " { A }
) }  -.  x  e. 
Singletons )
7573, 74sylibr 203 . . . . . . . 8  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( { ( F
" { A }
) }  i^i  Singletons )  =  (/) )
7675unieqd 3838 . . . . . . 7  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  ->  U. ( { ( F
" { A }
) }  i^i  Singletons )  =  U. (/) )
7776unieqd 3838 . . . . . 6  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  ->  U. U. ( { ( F " { A } ) }  i^i  Singletons )  =  U. U. (/) )
78 uni0 3854 . . . . . . . 8  |-  U. (/)  =  (/)
7978unieqi 3837 . . . . . . 7  |-  U. U. (/)  =  U. (/)
8079, 78eqtri 2303 . . . . . 6  |-  U. U. (/)  =  (/)
8177, 80syl6eq 2331 . . . . 5  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  ->  U. U. ( { ( F " { A } ) }  i^i  Singletons )  =  (/) )
825, 81syl5eq 2327 . . . 4  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( F `  A
)  =  (/) )
834, 82eqtr4d 2318 . . 3  |-  ( -.  A  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  -> 
( ( F  |`  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `  A
)  =  ( F `
 A ) )
843, 83pm2.61i 156 . 2  |-  ( ( F  |`  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) `
 A )  =  ( F `  A
)
852, 84eqtri 2303 1  |-  (Funpart F `  A )  =  ( F `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    i^i cin 3151   (/)c0 3455   {csn 3640   <.cop 3643   U.cuni 3827   class class class wbr 4023    X. cxp 4687   dom cdm 4689    |` cres 4691   "cima 4692    o. ccom 4693   ` cfv 5255  Singletoncsingle 24381   Singletonscsingles 24382  Imagecimage 24383  Funpartcfunpart 24392
This theorem is referenced by:  fullfunfv  24485
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-singleton 24403  df-singles 24404  df-image 24405  df-funpart 24415
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