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Theorem dfiota3 27954
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 5381 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 abeq1 2549 . . . . 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 1925 . . . . . 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 2975 . . . . . . . . 9  |-  y  e. 
_V
5 sneq 3887 . . . . . . . . . 10  |-  ( w  =  y  ->  { w }  =  { y } )
65eqeq2d 2454 . . . . . . . . 9  |-  ( w  =  y  ->  ( { x  |  ph }  =  { w }  <->  { x  |  ph }  =  {
y } ) )
74, 6ceqsexv 3009 . . . . . . . 8  |-  ( E. w ( w  =  y  /\  { x  |  ph }  =  {
w } )  <->  { x  |  ph }  =  {
y } )
8 snex 4533 . . . . . . . . . . 11  |-  { w }  e.  _V
9 eqeq1 2449 . . . . . . . . . . . . 13  |-  ( z  =  { w }  ->  ( z  =  {
x  |  ph }  <->  { w }  =  {
x  |  ph }
) )
10 eleq2 2504 . . . . . . . . . . . . 13  |-  ( z  =  { w }  ->  ( y  e.  z  <-> 
y  e.  { w } ) )
119, 10anbi12d 710 . . . . . . . . . . . 12  |-  ( z  =  { w }  ->  ( ( z  =  { x  |  ph }  /\  y  e.  z )  <->  ( { w }  =  { x  |  ph }  /\  y  e.  { w } ) ) )
12 eqcom 2445 . . . . . . . . . . . . 13  |-  ( { w }  =  {
x  |  ph }  <->  { x  |  ph }  =  { w } )
13 elsn 3891 . . . . . . . . . . . . . 14  |-  ( y  e.  { w }  <->  y  =  w )
14 equcom 1732 . . . . . . . . . . . . . 14  |-  ( y  =  w  <->  w  =  y )
1513, 14bitri 249 . . . . . . . . . . . . 13  |-  ( y  e.  { w }  <->  w  =  y )
1612, 15anbi12ci 698 . . . . . . . . . . . 12  |-  ( ( { w }  =  { x  |  ph }  /\  y  e.  { w } )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) )
1711, 16syl6bb 261 . . . . . . . . . . 11  |-  ( z  =  { w }  ->  ( ( z  =  { x  |  ph }  /\  y  e.  z )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) ) )
188, 17ceqsexv 3009 . . . . . . . . . 10  |-  ( E. z ( z  =  { w }  /\  ( z  =  {
x  |  ph }  /\  y  e.  z
) )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) )
19 an13 797 . . . . . . . . . . 11  |-  ( ( z  =  { w }  /\  ( z  =  { x  |  ph }  /\  y  e.  z ) )  <->  ( y  e.  z  /\  (
z  =  { x  |  ph }  /\  z  =  { w } ) ) )
2019exbii 1634 . . . . . . . . . 10  |-  ( E. z ( z  =  { w }  /\  ( z  =  {
x  |  ph }  /\  y  e.  z
) )  <->  E. z
( y  e.  z  /\  ( z  =  { x  |  ph }  /\  z  =  {
w } ) ) )
2118, 20bitr3i 251 . . . . . . . . 9  |-  ( ( w  =  y  /\  { x  |  ph }  =  { w } )  <->  E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
2221exbii 1634 . . . . . . . 8  |-  ( E. w ( w  =  y  /\  { x  |  ph }  =  {
w } )  <->  E. w E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
237, 22bitr3i 251 . . . . . . 7  |-  ( { x  |  ph }  =  { y }  <->  E. w E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
24 excom 1787 . . . . . . 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 249 . . . . . 6  |-  ( { x  |  ph }  =  { y }  <->  E. z E. w ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
26 eluniab 4102 . . . . . 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 277 . . . . 5  |-  ( { x  |  ph }  =  { y }  <->  y  e.  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) } )
282, 27mpgbir 1595 . . . 4  |-  { y  |  { x  | 
ph }  =  {
y } }  =  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) }
29 df-sn 3878 . . . . . . 7  |-  { {
x  |  ph } }  =  { z  |  z  =  {
x  |  ph } }
30 dfsingles2 27952 . . . . . . 7  |-  Singletons  =  {
z  |  E. w  z  =  { w } }
3129, 30ineq12i 3550 . . . . . 6  |-  ( { { x  |  ph } }  i^i  Singletons )  =  ( { z  |  z  =  { x  | 
ph } }  i^i  { z  |  E. w  z  =  { w } } )
32 inab 3618 . . . . . . 7  |-  ( { z  |  z  =  { x  |  ph } }  i^i  { z  |  E. w  z  =  { w } } )  =  {
z  |  ( z  =  { x  | 
ph }  /\  E. w  z  =  {
w } ) }
33 19.42v 1924 . . . . . . . . 9  |-  ( E. w ( z  =  { x  |  ph }  /\  z  =  {
w } )  <->  ( z  =  { x  |  ph }  /\  E. w  z  =  { w }
) )
3433bicomi 202 . . . . . . . 8  |-  ( ( z  =  { x  |  ph }  /\  E. w  z  =  {
w } )  <->  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) )
3534abbii 2555 . . . . . . 7  |-  { z  |  ( z  =  { x  |  ph }  /\  E. w  z  =  { w }
) }  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3632, 35eqtri 2463 . . . . . 6  |-  ( { z  |  z  =  { x  |  ph } }  i^i  { z  |  E. w  z  =  { w } } )  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3731, 36eqtri 2463 . . . . 5  |-  ( { { x  |  ph } }  i^i  Singletons )  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3837unieqi 4100 . . . 4  |-  U. ( { { x  |  ph } }  i^i  Singletons )  =  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) }
3928, 38eqtr4i 2466 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  U. ( { { x  |  ph } }  i^i  Singletons )
4039unieqi 4100 . 2  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. U. ( { {
x  |  ph } }  i^i  Singletons )
411, 40eqtri 2463 1  |-  ( iota
x ph )  =  U. U. ( { { x  |  ph } }  i^i  Singletons )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429    i^i cin 3327   {csn 3877   U.cuni 4091   iotacio 5379   Singletonscsingles 27869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-eprel 4632  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424  df-fv 5426  df-1st 6577  df-2nd 6578  df-symdif 27849  df-txp 27884  df-singleton 27892  df-singles 27893
This theorem is referenced by:  dffv5  27955
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