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Theorem funpartlem 29567
Description: Lemma for funpartfun 29568. 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 3104 . 2  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  ->  A  e.  _V )
2 ssnid 4043 . . . . 5  |-  x  e. 
{ x }
3 eleq2 2516 . . . . 5  |-  ( ( F " { A } )  =  {
x }  ->  (
x  e.  ( F
" { A }
)  <->  x  e.  { x } ) )
42, 3mpbiri 233 . . . 4  |-  ( ( F " { A } )  =  {
x }  ->  x  e.  ( F " { A } ) )
5 n0i 3775 . . . . 5  |-  ( x  e.  ( F " { A } )  ->  -.  ( F " { A } )  =  (/) )
6 snprc 4078 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 194 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
87imaeq2d 5327 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  ( F " (/) ) )
9 ima0 5342 . . . . . 6  |-  ( F
" (/) )  =  (/)
108, 9syl6eq 2500 . . . . 5  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  (/) )
115, 10nsyl2 127 . . . 4  |-  ( x  e.  ( F " { A } )  ->  A  e.  _V )
124, 11syl 16 . . 3  |-  ( ( F " { A } )  =  {
x }  ->  A  e.  _V )
1312exlimiv 1709 . 2  |-  ( E. x ( F " { A } )  =  { x }  ->  A  e.  _V )
14 eleq1 2515 . . 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 4024 . . . . . 6  |-  ( y  =  A  ->  { y }  =  { A } )
1615imaeq2d 5327 . . . . 5  |-  ( y  =  A  ->  ( F " { y } )  =  ( F
" { A }
) )
1716eqeq1d 2445 . . . 4  |-  ( y  =  A  ->  (
( F " {
y } )  =  { x }  <->  ( F " { A } )  =  { x }
) )
1817exbidv 1701 . . 3  |-  ( y  =  A  ->  ( E. x ( F " { y } )  =  { x }  <->  E. x ( F " { A } )  =  { x } ) )
19 vex 3098 . . . . 5  |-  y  e. 
_V
2019eldm 5190 . . . 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 5020 . . . . . . . . . 10  |-  ( y ( _V  X.  Singletons ) z  <->  ( y  e.  _V  /\  z  e.  Singletons
) )
2219, 21mpbiran 918 . . . . . . . . 9  |-  ( y ( _V  X.  Singletons ) z  <->  z  e.  Singletons )
23 elsingles 29543 . . . . . . . . 9  |-  ( z  e.  Singletons 
<->  E. x  z  =  { x } )
2422, 23bitri 249 . . . . . . . 8  |-  ( y ( _V  X.  Singletons ) z  <->  E. x  z  =  { x } )
2524anbi2i 694 . . . . . . 7  |-  ( ( y (Image F  o. Singleton ) z  /\  y ( _V  X.  Singletons ) z )  <-> 
( y (Image F  o. Singleton ) z  /\  E. x  z  =  {
x } ) )
26 brin 4486 . . . . . . 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 1761 . . . . . . 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 1654 . . . . 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 1835 . . . . 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 1658 . . . . . 6  |-  ( E. z ( y (Image
F  o. Singleton ) z  /\  z  =  { x } )  <->  E. z
( z  =  {
x }  /\  y
(Image F  o. Singleton ) z ) )
33 snex 4678 . . . . . . 7  |-  { x }  e.  _V
34 breq2 4441 . . . . . . 7  |-  ( z  =  { x }  ->  ( y (Image F  o. Singleton ) z  <->  y (Image F  o. Singleton ) { x } ) )
3533, 34ceqsexv 3132 . . . . . 6  |-  ( E. z ( z  =  { x }  /\  y (Image F  o. Singleton ) z )  <->  y (Image F  o. Singleton ) { x }
)
3619, 33brco 5163 . . . . . . 7  |-  ( y (Image F  o. Singleton ) { x }  <->  E. z
( ySingleton z  /\  zImage F { x } ) )
37 vex 3098 . . . . . . . . . 10  |-  z  e. 
_V
3819, 37brsingle 29542 . . . . . . . . 9  |-  ( ySingleton
z  <->  z  =  {
y } )
3938anbi1i 695 . . . . . . . 8  |-  ( ( ySingleton z  /\  zImage F { x } )  <-> 
( z  =  {
y }  /\  zImage F { x } ) )
4039exbii 1654 . . . . . . 7  |-  ( E. z ( ySingleton z  /\  zImage F { x } )  <->  E. z
( z  =  {
y }  /\  zImage F { x } ) )
41 snex 4678 . . . . . . . . 9  |-  { y }  e.  _V
42 breq1 4440 . . . . . . . . 9  |-  ( z  =  { y }  ->  ( zImage F { x }  <->  { y }Image F { x }
) )
4341, 42ceqsexv 3132 . . . . . . . 8  |-  ( E. z ( z  =  { y }  /\  zImage F { x }
)  <->  { y }Image F { x } )
4441, 33brimage 29551 . . . . . . . 8  |-  ( { y }Image F {
x }  <->  { x }  =  ( F " { y } ) )
45 eqcom 2452 . . . . . . . 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 1654 . . . 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 3154 . 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 353 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 369    = wceq 1383   E.wex 1599    e. wcel 1804   _Vcvv 3095    i^i cin 3460   (/)c0 3770   {csn 4014   class class class wbr 4437    X. cxp 4987   dom cdm 4989   "cima 4992    o. ccom 4993  Singletoncsingle 29462   Singletonscsingles 29463  Imagecimage 29464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-eprel 4781  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-1st 6785  df-2nd 6786  df-symdif 29443  df-txp 29478  df-singleton 29486  df-singles 29487  df-image 29488
This theorem is referenced by:  funpartfun  29568  funpartfv  29570
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