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Theorem funpartlem 30714
Description: Lemma for funpartfun 30715. 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 3089 . 2  |-  ( A  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  ->  A  e.  _V )
2 ssnid 4027 . . . . 5  |-  x  e. 
{ x }
3 eleq2 2496 . . . . 5  |-  ( ( F " { A } )  =  {
x }  ->  (
x  e.  ( F
" { A }
)  <->  x  e.  { x } ) )
42, 3mpbiri 236 . . . 4  |-  ( ( F " { A } )  =  {
x }  ->  x  e.  ( F " { A } ) )
5 n0i 3766 . . . . 5  |-  ( x  e.  ( F " { A } )  ->  -.  ( F " { A } )  =  (/) )
6 snprc 4063 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
76biimpi 197 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
87imaeq2d 5187 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  ( F " (/) ) )
9 ima0 5202 . . . . . 6  |-  ( F
" (/) )  =  (/)
108, 9syl6eq 2479 . . . . 5  |-  ( -.  A  e.  _V  ->  ( F " { A } )  =  (/) )
115, 10nsyl2 130 . . . 4  |-  ( x  e.  ( F " { A } )  ->  A  e.  _V )
124, 11syl 17 . . 3  |-  ( ( F " { A } )  =  {
x }  ->  A  e.  _V )
1312exlimiv 1770 . 2  |-  ( E. x ( F " { A } )  =  { x }  ->  A  e.  _V )
14 eleq1 2495 . . 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 4008 . . . . . 6  |-  ( y  =  A  ->  { y }  =  { A } )
1615imaeq2d 5187 . . . . 5  |-  ( y  =  A  ->  ( F " { y } )  =  ( F
" { A }
) )
1716eqeq1d 2424 . . . 4  |-  ( y  =  A  ->  (
( F " {
y } )  =  { x }  <->  ( F " { A } )  =  { x }
) )
1817exbidv 1762 . . 3  |-  ( y  =  A  ->  ( E. x ( F " { y } )  =  { x }  <->  E. x ( F " { A } )  =  { x } ) )
19 vex 3083 . . . . 5  |-  y  e. 
_V
2019eldm 5051 . . . 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 4884 . . . . . . . . . 10  |-  ( y ( _V  X.  Singletons ) z  <->  ( y  e.  _V  /\  z  e.  Singletons
) )
2219, 21mpbiran 926 . . . . . . . . 9  |-  ( y ( _V  X.  Singletons ) z  <->  z  e.  Singletons )
23 elsingles 30690 . . . . . . . . 9  |-  ( z  e.  Singletons 
<->  E. x  z  =  { x } )
2422, 23bitri 252 . . . . . . . 8  |-  ( y ( _V  X.  Singletons ) z  <->  E. x  z  =  { x } )
2524anbi2i 698 . . . . . . 7  |-  ( ( y (Image F  o. Singleton ) z  /\  y ( _V  X.  Singletons ) z )  <-> 
( y (Image F  o. Singleton ) z  /\  E. x  z  =  {
x } ) )
26 brin 4473 . . . . . . 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 1827 . . . . . . 7  |-  ( E. x ( y (Image
F  o. Singleton ) z  /\  z  =  { x } )  <->  ( y
(Image F  o. Singleton ) z  /\  E. x  z  =  { x }
) )
2825, 26, 273bitr4i 280 . . . . . 6  |-  ( y ( (Image F  o. Singleton )  i^i  ( _V  X.  Singletons ) ) z  <->  E. x
( y (Image F  o. Singleton ) z  /\  z  =  { x } ) )
2928exbii 1712 . . . . 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 1903 . . . . 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 252 . . . 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 1716 . . . . . 6  |-  ( E. z ( y (Image
F  o. Singleton ) z  /\  z  =  { x } )  <->  E. z
( z  =  {
x }  /\  y
(Image F  o. Singleton ) z ) )
33 snex 4662 . . . . . . 7  |-  { x }  e.  _V
34 breq2 4427 . . . . . . 7  |-  ( z  =  { x }  ->  ( y (Image F  o. Singleton ) z  <->  y (Image F  o. Singleton ) { x } ) )
3533, 34ceqsexv 3118 . . . . . 6  |-  ( E. z ( z  =  { x }  /\  y (Image F  o. Singleton ) z )  <->  y (Image F  o. Singleton ) { x }
)
3619, 33brco 5024 . . . . . . 7  |-  ( y (Image F  o. Singleton ) { x }  <->  E. z
( ySingleton z  /\  zImage F { x } ) )
37 vex 3083 . . . . . . . . . 10  |-  z  e. 
_V
3819, 37brsingle 30689 . . . . . . . . 9  |-  ( ySingleton
z  <->  z  =  {
y } )
3938anbi1i 699 . . . . . . . 8  |-  ( ( ySingleton z  /\  zImage F { x } )  <-> 
( z  =  {
y }  /\  zImage F { x } ) )
4039exbii 1712 . . . . . . 7  |-  ( E. z ( ySingleton z  /\  zImage F { x } )  <->  E. z
( z  =  {
y }  /\  zImage F { x } ) )
41 snex 4662 . . . . . . . . 9  |-  { y }  e.  _V
42 breq1 4426 . . . . . . . . 9  |-  ( z  =  { y }  ->  ( zImage F { x }  <->  { y }Image F { x }
) )
4341, 42ceqsexv 3118 . . . . . . . 8  |-  ( E. z ( z  =  { y }  /\  zImage F { x }
)  <->  { y }Image F { x } )
4441, 33brimage 30698 . . . . . . . 8  |-  ( { y }Image F {
x }  <->  { x }  =  ( F " { y } ) )
45 eqcom 2431 . . . . . . . 8  |-  ( { x }  =  ( F " { y } )  <->  ( F " { y } )  =  { x }
)
4643, 44, 453bitri 274 . . . . . . 7  |-  ( E. z ( z  =  { y }  /\  zImage F { x }
)  <->  ( F " { y } )  =  { x }
)
4736, 40, 463bitri 274 . . . . . 6  |-  ( y (Image F  o. Singleton ) { x }  <->  ( F " { y } )  =  { x }
)
4832, 35, 473bitri 274 . . . . 5  |-  ( E. z ( y (Image
F  o. Singleton ) z  /\  z  =  { x } )  <->  ( F " { y } )  =  { x }
)
4948exbii 1712 . . . 4  |-  ( E. x E. z ( y (Image F  o. Singleton ) z  /\  z  =  { x } )  <->  E. x ( F " { y } )  =  { x }
)
5020, 31, 493bitri 274 . . 3  |-  ( y  e.  dom  ( (Image
F  o. Singleton )  i^i  ( _V  X.  Singletons ) )  <->  E. x
( F " {
y } )  =  { x } )
5114, 18, 50vtoclbg 3140 . 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 354 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 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   _Vcvv 3080    i^i cin 3435   (/)c0 3761   {csn 3998   class class class wbr 4423    X. cxp 4851   dom cdm 4853   "cima 4856    o. ccom 4857  Singletoncsingle 30609   Singletonscsingles 30610  Imagecimage 30611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-symdif 3693  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-eprel 4764  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-1st 6807  df-2nd 6808  df-txp 30625  df-singleton 30633  df-singles 30634  df-image 30635
This theorem is referenced by:  funpartfun  30715  funpartfv  30717
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