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Theorem funpartlem 30248
Description: Lemma for funpartfun 30249. Show membership in the restriction. (Contributed by Scott Fenton, 4-Dec-2017.)
Assertion
Ref Expression
funpartlem  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x
( F " { A } )  =  {
x } )
Distinct variable groups:    x, A    x, F

Proof of Theorem funpartlem
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3065 . 2  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  ->  A  e.  _V )
2 ssnid 3998 . . . . 5  |-  x  e. 
{ x }
3 eleq2 2473 . . . . 5  |-  ( ( F " { A } )  =  {
x }  ->  (
x  e.  ( F
" { A }
)  <->  x  e.  { x } ) )
42, 3mpbiri 233 . . . 4  |-  ( ( F " { A } )  =  {
x }  ->  x  e.  ( F " { A } ) )
5 n0i 3740 . . . . 5  |-  ( x  e.  ( F " { A } )  ->  -.  ( F " { A } )  =  (/) )
6 snprc 4032 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 194 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
87imaeq2d 5276 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  ( F " (/) ) )
9 ima0 5291 . . . . . 6  |-  ( F
" (/) )  =  (/)
108, 9syl6eq 2457 . . . . 5  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  (/) )
115, 10nsyl2 127 . . . 4  |-  ( x  e.  ( F " { A } )  ->  A  e.  _V )
124, 11syl 17 . . 3  |-  ( ( F " { A } )  =  {
x }  ->  A  e.  _V )
1312exlimiv 1741 . 2  |-  ( E. x ( F " { A } )  =  { x }  ->  A  e.  _V )
14 eleq1 2472 . . 3  |-  ( y  =  A  ->  (
y  e.  dom  (
(Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  A  e.  dom  ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) ) )
15 sneq 3979 . . . . . 6  |-  ( y  =  A  ->  { y }  =  { A } )
1615imaeq2d 5276 . . . . 5  |-  ( y  =  A  ->  ( F " { y } )  =  ( F
" { A }
) )
1716eqeq1d 2402 . . . 4  |-  ( y  =  A  ->  (
( F " {
y } )  =  { x }  <->  ( F " { A } )  =  { x }
) )
1817exbidv 1733 . . 3  |-  ( y  =  A  ->  ( E. x ( F " { y } )  =  { x }  <->  E. x ( F " { A } )  =  { x } ) )
19 vex 3059 . . . . 5  |-  y  e. 
_V
2019eldm 5140 . . . 4  |-  ( y  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. z 
y ( (Image F  o. Singleton )  i^i  ( _V 
X.  Singletons ) ) z )
21 brxp 4971 . . . . . . . . . 10  |-  ( y ( _V  X.  Singletons ) z  <->  ( y  e.  _V  /\  z  e.  Singletons
) )
2219, 21mpbiran 917 . . . . . . . . 9  |-  ( y ( _V  X.  Singletons ) z  <->  z  e.  Singletons )
23 elsingles 30224 . . . . . . . . 9  |-  ( z  e.  Singletons 
<->  E. x  z  =  { x } )
2422, 23bitri 249 . . . . . . . 8  |-  ( y ( _V  X.  Singletons ) z  <->  E. x  z  =  { x } )
2524anbi2i 692 . . . . . . 7  |-  ( ( y (Image F  o. Singleton ) z  /\  y ( _V  X.  Singletons ) z )  <-> 
( y (Image F  o. Singleton ) z  /\  E. x  z  =  {
x } ) )
26 brin 4441 . . . . . . 7  |-  ( y ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) z  <->  ( y
(Image F  o. Singleton ) z  /\  y ( _V 
X.  Singletons ) z ) )
27 19.42v 1797 . . . . . . 7  |-  ( E. x ( y (Image
F  o. Singleton ) z  /\  z  =  { x } )  <->  ( y
(Image F  o. Singleton ) z  /\  E. x  z  =  { x }
) )
2825, 26, 273bitr4i 277 . . . . . 6  |-  ( y ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) z  <->  E. x
( y (Image F  o. Singleton ) z  /\  z  =  { x } ) )
2928exbii 1686 . . . . 5  |-  ( E. z  y ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) z  <->  E. z E. x ( y (Image F  o. Singleton ) z  /\  z  =  { x } ) )
30 excom 1871 . . . . 5  |-  ( E. z E. x ( y (Image F  o. Singleton ) z  /\  z  =  { x } )  <->  E. x E. z ( y (Image F  o. Singleton ) z  /\  z  =  { x } ) )
3129, 30bitri 249 . . . 4  |-  ( E. z  y ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) z  <->  E. x E. z ( y (Image F  o. Singleton ) z  /\  z  =  { x } ) )
32 exancom 1690 . . . . . 6  |-  ( E. z ( y (Image
F  o. Singleton ) z  /\  z  =  { x } )  <->  E. z
( z  =  {
x }  /\  y
(Image F  o. Singleton ) z ) )
33 snex 4629 . . . . . . 7  |-  { x }  e.  _V
34 breq2 4396 . . . . . . 7  |-  ( z  =  { x }  ->  ( y (Image F  o. Singleton ) z  <->  y (Image F  o. Singleton ) { x } ) )
3533, 34ceqsexv 3093 . . . . . 6  |-  ( E. z ( z  =  { x }  /\  y (Image F  o. Singleton ) z )  <->  y (Image F  o. Singleton ) { x }
)
3619, 33brco 5113 . . . . . . 7  |-  ( y (Image F  o. Singleton ) { x }  <->  E. z
( ySingleton z  /\  zImage F { x } ) )
37 vex 3059 . . . . . . . . . 10  |-  z  e. 
_V
3819, 37brsingle 30223 . . . . . . . . 9  |-  ( ySingleton
z  <->  z  =  {
y } )
3938anbi1i 693 . . . . . . . 8  |-  ( ( ySingleton z  /\  zImage F { x } )  <-> 
( z  =  {
y }  /\  zImage F { x } ) )
4039exbii 1686 . . . . . . 7  |-  ( E. z ( ySingleton z  /\  zImage F { x } )  <->  E. z
( z  =  {
y }  /\  zImage F { x } ) )
41 snex 4629 . . . . . . . . 9  |-  { y }  e.  _V
42 breq1 4395 . . . . . . . . 9  |-  ( z  =  { y }  ->  ( zImage F { x }  <->  { y }Image F { x }
) )
4341, 42ceqsexv 3093 . . . . . . . 8  |-  ( E. z ( z  =  { y }  /\  zImage F { x }
)  <->  { y }Image F { x } )
4441, 33brimage 30232 . . . . . . . 8  |-  ( { y }Image F {
x }  <->  { x }  =  ( F " { y } ) )
45 eqcom 2409 . . . . . . . 8  |-  ( { x }  =  ( F " { y } )  <->  ( F " { y } )  =  { x }
)
4643, 44, 453bitri 271 . . . . . . 7  |-  ( E. z ( z  =  { y }  /\  zImage F { x }
)  <->  ( F " { y } )  =  { x }
)
4736, 40, 463bitri 271 . . . . . 6  |-  ( y (Image F  o. Singleton ) { x }  <->  ( F " { y } )  =  { x }
)
4832, 35, 473bitri 271 . . . . 5  |-  ( E. z ( y (Image
F  o. Singleton ) z  /\  z  =  { x } )  <->  ( F " { y } )  =  { x }
)
4948exbii 1686 . . . 4  |-  ( E. x E. z ( y (Image F  o. Singleton ) z  /\  z  =  { x } )  <->  E. x ( F " { y } )  =  { x }
)
5020, 31, 493bitri 271 . . 3  |-  ( y  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x
( F " {
y } )  =  { x } )
5114, 18, 50vtoclbg 3115 . 2  |-  ( A  e.  _V  ->  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x
( F " { A } )  =  {
x } ) )
521, 13, 51pm5.21nii 351 1  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x
( F " { A } )  =  {
x } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367    = wceq 1403   E.wex 1631    e. wcel 1840   _Vcvv 3056    i^i cin 3410   (/)c0 3735   {csn 3969   class class class wbr 4392    X. cxp 4938   dom cdm 4940   "cima 4943    o. ccom 4944  Singletoncsingle 30143   Singletonscsingles 30144  Imagecimage 30145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-symdif 3667  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-eprel 4731  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fo 5529  df-fv 5531  df-1st 6736  df-2nd 6737  df-txp 30159  df-singleton 30167  df-singles 30168  df-image 30169
This theorem is referenced by:  funpartfun  30249  funpartfv  30251
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