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Theorem dfiota3 30632
Description: A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.)
Assertion
Ref Expression
dfiota3  |-  ( iota
x ph )  =  U. U. ( { { x  |  ph } }  i^i  Singletons )

Proof of Theorem dfiota3
Dummy variables  y 
z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iota 5501 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 abeq1 2532 . . . . 5  |-  ( { y  |  { x  |  ph }  =  {
y } }  =  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) }  <->  A. y ( { x  |  ph }  =  {
y }  <->  y  e.  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) } ) )
3 exdistr 1828 . . . . . 6  |-  ( E. z E. w ( y  e.  z  /\  ( z  =  {
x  |  ph }  /\  z  =  {
w } ) )  <->  E. z ( y  e.  z  /\  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) ) )
4 vex 3019 . . . . . . . . 9  |-  y  e. 
_V
5 sneq 3944 . . . . . . . . . 10  |-  ( w  =  y  ->  { w }  =  { y } )
65eqeq2d 2432 . . . . . . . . 9  |-  ( w  =  y  ->  ( { x  |  ph }  =  { w }  <->  { x  |  ph }  =  {
y } ) )
74, 6ceqsexv 3054 . . . . . . . 8  |-  ( E. w ( w  =  y  /\  { x  |  ph }  =  {
w } )  <->  { x  |  ph }  =  {
y } )
8 snex 4598 . . . . . . . . . . 11  |-  { w }  e.  _V
9 eqeq1 2426 . . . . . . . . . . . . 13  |-  ( z  =  { w }  ->  ( z  =  {
x  |  ph }  <->  { w }  =  {
x  |  ph }
) )
10 eleq2 2489 . . . . . . . . . . . . 13  |-  ( z  =  { w }  ->  ( y  e.  z  <-> 
y  e.  { w } ) )
119, 10anbi12d 715 . . . . . . . . . . . 12  |-  ( z  =  { w }  ->  ( ( z  =  { x  |  ph }  /\  y  e.  z )  <->  ( { w }  =  { x  |  ph }  /\  y  e.  { w } ) ) )
12 eqcom 2429 . . . . . . . . . . . . 13  |-  ( { w }  =  {
x  |  ph }  <->  { x  |  ph }  =  { w } )
13 elsn 3948 . . . . . . . . . . . . . 14  |-  ( y  e.  { w }  <->  y  =  w )
14 equcom 1848 . . . . . . . . . . . . . 14  |-  ( y  =  w  <->  w  =  y )
1513, 14bitri 252 . . . . . . . . . . . . 13  |-  ( y  e.  { w }  <->  w  =  y )
1612, 15anbi12ci 702 . . . . . . . . . . . 12  |-  ( ( { w }  =  { x  |  ph }  /\  y  e.  { w } )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) )
1711, 16syl6bb 264 . . . . . . . . . . 11  |-  ( z  =  { w }  ->  ( ( z  =  { x  |  ph }  /\  y  e.  z )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) ) )
188, 17ceqsexv 3054 . . . . . . . . . 10  |-  ( E. z ( z  =  { w }  /\  ( z  =  {
x  |  ph }  /\  y  e.  z
) )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) )
19 an13 806 . . . . . . . . . . 11  |-  ( ( z  =  { w }  /\  ( z  =  { x  |  ph }  /\  y  e.  z ) )  <->  ( y  e.  z  /\  (
z  =  { x  |  ph }  /\  z  =  { w } ) ) )
2019exbii 1712 . . . . . . . . . 10  |-  ( E. z ( z  =  { w }  /\  ( z  =  {
x  |  ph }  /\  y  e.  z
) )  <->  E. z
( y  e.  z  /\  ( z  =  { x  |  ph }  /\  z  =  {
w } ) ) )
2118, 20bitr3i 254 . . . . . . . . 9  |-  ( ( w  =  y  /\  { x  |  ph }  =  { w } )  <->  E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
2221exbii 1712 . . . . . . . 8  |-  ( E. w ( w  =  y  /\  { x  |  ph }  =  {
w } )  <->  E. w E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
237, 22bitr3i 254 . . . . . . 7  |-  ( { x  |  ph }  =  { y }  <->  E. w E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
24 excom 1903 . . . . . . 7  |-  ( E. w E. z ( y  e.  z  /\  ( z  =  {
x  |  ph }  /\  z  =  {
w } ) )  <->  E. z E. w ( y  e.  z  /\  ( z  =  {
x  |  ph }  /\  z  =  {
w } ) ) )
2523, 24bitri 252 . . . . . 6  |-  ( { x  |  ph }  =  { y }  <->  E. z E. w ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
26 eluniab 4166 . . . . . 6  |-  ( y  e.  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  { w } ) }  <->  E. z ( y  e.  z  /\  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) ) )
273, 25, 263bitr4i 280 . . . . 5  |-  ( { x  |  ph }  =  { y }  <->  y  e.  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) } )
282, 27mpgbir 1667 . . . 4  |-  { y  |  { x  | 
ph }  =  {
y } }  =  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) }
29 df-sn 3935 . . . . . . 7  |-  { {
x  |  ph } }  =  { z  |  z  =  {
x  |  ph } }
30 dfsingles2 30630 . . . . . . 7  |-  Singletons  =  {
z  |  E. w  z  =  { w } }
3129, 30ineq12i 3598 . . . . . 6  |-  ( { { x  |  ph } }  i^i  Singletons )  =  ( { z  |  z  =  { x  | 
ph } }  i^i  { z  |  E. w  z  =  { w } } )
32 inab 3677 . . . . . . 7  |-  ( { z  |  z  =  { x  |  ph } }  i^i  { z  |  E. w  z  =  { w } } )  =  {
z  |  ( z  =  { x  | 
ph }  /\  E. w  z  =  {
w } ) }
33 19.42v 1827 . . . . . . . . 9  |-  ( E. w ( z  =  { x  |  ph }  /\  z  =  {
w } )  <->  ( z  =  { x  |  ph }  /\  E. w  z  =  { w }
) )
3433bicomi 205 . . . . . . . 8  |-  ( ( z  =  { x  |  ph }  /\  E. w  z  =  {
w } )  <->  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) )
3534abbii 2538 . . . . . . 7  |-  { z  |  ( z  =  { x  |  ph }  /\  E. w  z  =  { w }
) }  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3632, 35eqtri 2444 . . . . . 6  |-  ( { z  |  z  =  { x  |  ph } }  i^i  { z  |  E. w  z  =  { w } } )  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3731, 36eqtri 2444 . . . . 5  |-  ( { { x  |  ph } }  i^i  Singletons )  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3837unieqi 4164 . . . 4  |-  U. ( { { x  |  ph } }  i^i  Singletons )  =  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) }
3928, 38eqtr4i 2447 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  U. ( { { x  |  ph } }  i^i  Singletons )
4039unieqi 4164 . 2  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. U. ( { {
x  |  ph } }  i^i  Singletons )
411, 40eqtri 2444 1  |-  ( iota
x ph )  =  U. U. ( { { x  |  ph } }  i^i  Singletons )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2408    i^i cin 3371   {csn 3934   U.cuni 4155   iotacio 5499   Singletonscsingles 30547
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 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534
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 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-rab 2717  df-v 3018  df-sbc 3236  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-symdif 3629  df-nul 3698  df-if 3848  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-br 4360  df-opab 4419  df-mpt 4420  df-eprel 4700  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-fo 5543  df-fv 5545  df-1st 6744  df-2nd 6745  df-txp 30562  df-singleton 30570  df-singles 30571
This theorem is referenced by:  dffv5  30633
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