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Theorem funpartlem 32049
Description: Lemma for funpartfun 32050. 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 3212 . 2  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  ->  A  e.  _V )
2 vsnid 4209 . . . . 5  |-  x  e. 
{ x }
3 eleq2 2690 . . . . 5  |-  ( ( F " { A } )  =  {
x }  ->  (
x  e.  ( F
" { A }
)  <->  x  e.  { x } ) )
42, 3mpbiri 248 . . . 4  |-  ( ( F " { A } )  =  {
x }  ->  x  e.  ( F " { A } ) )
5 n0i 3920 . . . . 5  |-  ( x  e.  ( F " { A } )  ->  -.  ( F " { A } )  =  (/) )
6 snprc 4253 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 206 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
87imaeq2d 5466 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  ( F " (/) ) )
9 ima0 5481 . . . . . 6  |-  ( F
" (/) )  =  (/)
108, 9syl6eq 2672 . . . . 5  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  (/) )
115, 10nsyl2 142 . . . 4  |-  ( x  e.  ( F " { A } )  ->  A  e.  _V )
124, 11syl 17 . . 3  |-  ( ( F " { A } )  =  {
x }  ->  A  e.  _V )
1312exlimiv 1858 . 2  |-  ( E. x ( F " { A } )  =  { x }  ->  A  e.  _V )
14 eleq1 2689 . . 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 4187 . . . . . 6  |-  ( y  =  A  ->  { y }  =  { A } )
1615imaeq2d 5466 . . . . 5  |-  ( y  =  A  ->  ( F " { y } )  =  ( F
" { A }
) )
1716eqeq1d 2624 . . . 4  |-  ( y  =  A  ->  (
( F " {
y } )  =  { x }  <->  ( F " { A } )  =  { x }
) )
1817exbidv 1850 . . 3  |-  ( y  =  A  ->  ( E. x ( F " { y } )  =  { x }  <->  E. x ( F " { A } )  =  { x } ) )
19 vex 3203 . . . . 5  |-  y  e. 
_V
2019eldm 5321 . . . 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 5147 . . . . . . . . . 10  |-  ( y ( _V  X.  Singletons ) z  <->  ( y  e.  _V  /\  z  e.  Singletons
) )
2219, 21mpbiran 953 . . . . . . . . 9  |-  ( y ( _V  X.  Singletons ) z  <->  z  e.  Singletons )
23 elsingles 32025 . . . . . . . . 9  |-  ( z  e.  Singletons 
<->  E. x  z  =  { x } )
2422, 23bitri 264 . . . . . . . 8  |-  ( y ( _V  X.  Singletons ) z  <->  E. x  z  =  { x } )
2524anbi2i 730 . . . . . . 7  |-  ( ( y (Image F  o. Singleton ) z  /\  y ( _V  X.  Singletons ) z )  <-> 
( y (Image F  o. Singleton ) z  /\  E. x  z  =  {
x } ) )
26 brin 4704 . . . . . . 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 1918 . . . . . . 7  |-  ( E. x ( y (Image
F  o. Singleton ) z  /\  z  =  { x } )  <->  ( y
(Image F  o. Singleton ) z  /\  E. x  z  =  { x }
) )
2825, 26, 273bitr4i 292 . . . . . 6  |-  ( y ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) z  <->  E. x
( y (Image F  o. Singleton ) z  /\  z  =  { x } ) )
2928exbii 1774 . . . . 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 2042 . . . . 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 264 . . . 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 1787 . . . . . 6  |-  ( E. z ( y (Image
F  o. Singleton ) z  /\  z  =  { x } )  <->  E. z
( z  =  {
x }  /\  y
(Image F  o. Singleton ) z ) )
33 snex 4908 . . . . . . 7  |-  { x }  e.  _V
34 breq2 4657 . . . . . . 7  |-  ( z  =  { x }  ->  ( y (Image F  o. Singleton ) z  <->  y (Image F  o. Singleton ) { x } ) )
3533, 34ceqsexv 3242 . . . . . 6  |-  ( E. z ( z  =  { x }  /\  y (Image F  o. Singleton ) z )  <->  y (Image F  o. Singleton ) { x }
)
3619, 33brco 5292 . . . . . . 7  |-  ( y (Image F  o. Singleton ) { x }  <->  E. z
( ySingleton z  /\  zImage F { x } ) )
37 vex 3203 . . . . . . . . . 10  |-  z  e. 
_V
3819, 37brsingle 32024 . . . . . . . . 9  |-  ( ySingleton
z  <->  z  =  {
y } )
3938anbi1i 731 . . . . . . . 8  |-  ( ( ySingleton z  /\  zImage F { x } )  <-> 
( z  =  {
y }  /\  zImage F { x } ) )
4039exbii 1774 . . . . . . 7  |-  ( E. z ( ySingleton z  /\  zImage F { x } )  <->  E. z
( z  =  {
y }  /\  zImage F { x } ) )
41 snex 4908 . . . . . . . . 9  |-  { y }  e.  _V
42 breq1 4656 . . . . . . . . 9  |-  ( z  =  { y }  ->  ( zImage F { x }  <->  { y }Image F { x }
) )
4341, 42ceqsexv 3242 . . . . . . . 8  |-  ( E. z ( z  =  { y }  /\  zImage F { x }
)  <->  { y }Image F { x } )
4441, 33brimage 32033 . . . . . . . 8  |-  ( { y }Image F {
x }  <->  { x }  =  ( F " { y } ) )
45 eqcom 2629 . . . . . . . 8  |-  ( { x }  =  ( F " { y } )  <->  ( F " { y } )  =  { x }
)
4643, 44, 453bitri 286 . . . . . . 7  |-  ( E. z ( z  =  { y }  /\  zImage F { x }
)  <->  ( F " { y } )  =  { x }
)
4736, 40, 463bitri 286 . . . . . 6  |-  ( y (Image F  o. Singleton ) { x }  <->  ( F " { y } )  =  { x }
)
4832, 35, 473bitri 286 . . . . 5  |-  ( E. z ( y (Image
F  o. Singleton ) z  /\  z  =  { x } )  <->  ( F " { y } )  =  { x }
)
4948exbii 1774 . . . 4  |-  ( E. x E. z ( y (Image F  o. Singleton ) z  /\  z  =  { x } )  <->  E. x ( F " { y } )  =  { x }
)
5020, 31, 493bitri 286 . . 3  |-  ( y  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x
( F " {
y } )  =  { x } )
5114, 18, 50vtoclbg 3267 . 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 368 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 196    /\ wa 384    = wceq 1483   E.wex 1704    e. wcel 1990   _Vcvv 3200    i^i cin 3573   (/)c0 3915   {csn 4177   class class class wbr 4653    X. cxp 5112   dom cdm 5114   "cima 5117    o. ccom 5118  Singletoncsingle 31945   Singletonscsingles 31946  Imagecimage 31947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-singleton 31969  df-singles 31970  df-image 31971
This theorem is referenced by:  funpartfun  32050  funpartfv  32052
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