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Theorem dfiota3 24533
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 5235 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 abeq1 2402 . . . . 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 1859 . . . . . 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 2804 . . . . . . . . 9  |-  y  e. 
_V
5 sneq 3664 . . . . . . . . . 10  |-  ( w  =  y  ->  { w }  =  { y } )
65eqeq2d 2307 . . . . . . . . 9  |-  ( w  =  y  ->  ( { x  |  ph }  =  { w }  <->  { x  |  ph }  =  {
y } ) )
74, 6ceqsexv 2836 . . . . . . . 8  |-  ( E. w ( w  =  y  /\  { x  |  ph }  =  {
w } )  <->  { x  |  ph }  =  {
y } )
8 snex 4232 . . . . . . . . . . 11  |-  { w }  e.  _V
9 eqeq1 2302 . . . . . . . . . . . . 13  |-  ( z  =  { w }  ->  ( z  =  {
x  |  ph }  <->  { w }  =  {
x  |  ph }
) )
10 eleq2 2357 . . . . . . . . . . . . 13  |-  ( z  =  { w }  ->  ( y  e.  z  <-> 
y  e.  { w } ) )
119, 10anbi12d 691 . . . . . . . . . . . 12  |-  ( z  =  { w }  ->  ( ( z  =  { x  |  ph }  /\  y  e.  z )  <->  ( { w }  =  { x  |  ph }  /\  y  e.  { w } ) ) )
12 eqcom 2298 . . . . . . . . . . . . 13  |-  ( { w }  =  {
x  |  ph }  <->  { x  |  ph }  =  { w } )
13 elsn 3668 . . . . . . . . . . . . . 14  |-  ( y  e.  { w }  <->  y  =  w )
14 equcom 1665 . . . . . . . . . . . . . 14  |-  ( y  =  w  <->  w  =  y )
1513, 14bitri 240 . . . . . . . . . . . . 13  |-  ( y  e.  { w }  <->  w  =  y )
1612, 15anbi12ci 679 . . . . . . . . . . . 12  |-  ( ( { w }  =  { x  |  ph }  /\  y  e.  { w } )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) )
1711, 16syl6bb 252 . . . . . . . . . . 11  |-  ( z  =  { w }  ->  ( ( z  =  { x  |  ph }  /\  y  e.  z )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) ) )
188, 17ceqsexv 2836 . . . . . . . . . 10  |-  ( E. z ( z  =  { w }  /\  ( z  =  {
x  |  ph }  /\  y  e.  z
) )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) )
19 an13 774 . . . . . . . . . . 11  |-  ( ( z  =  { w }  /\  ( z  =  { x  |  ph }  /\  y  e.  z ) )  <->  ( y  e.  z  /\  (
z  =  { x  |  ph }  /\  z  =  { w } ) ) )
2019exbii 1572 . . . . . . . . . 10  |-  ( E. z ( z  =  { w }  /\  ( z  =  {
x  |  ph }  /\  y  e.  z
) )  <->  E. z
( y  e.  z  /\  ( z  =  { x  |  ph }  /\  z  =  {
w } ) ) )
2118, 20bitr3i 242 . . . . . . . . 9  |-  ( ( w  =  y  /\  { x  |  ph }  =  { w } )  <->  E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
2221exbii 1572 . . . . . . . 8  |-  ( E. w ( w  =  y  /\  { x  |  ph }  =  {
w } )  <->  E. w E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
237, 22bitr3i 242 . . . . . . 7  |-  ( { x  |  ph }  =  { y }  <->  E. w E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
24 excom 1798 . . . . . . 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 240 . . . . . 6  |-  ( { x  |  ph }  =  { y }  <->  E. z E. w ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
26 eluniab 3855 . . . . . 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 268 . . . . 5  |-  ( { x  |  ph }  =  { y }  <->  y  e.  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) } )
282, 27mpgbir 1540 . . . 4  |-  { y  |  { x  | 
ph }  =  {
y } }  =  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) }
29 df-sn 3659 . . . . . . 7  |-  { {
x  |  ph } }  =  { z  |  z  =  {
x  |  ph } }
30 dfsingles2 24531 . . . . . . 7  |-  Singletons  =  {
z  |  E. w  z  =  { w } }
3129, 30ineq12i 3381 . . . . . 6  |-  ( { { x  |  ph } }  i^i  Singletons )  =  ( { z  |  z  =  { x  | 
ph } }  i^i  { z  |  E. w  z  =  { w } } )
32 inab 3449 . . . . . . 7  |-  ( { z  |  z  =  { x  |  ph } }  i^i  { z  |  E. w  z  =  { w } } )  =  {
z  |  ( z  =  { x  | 
ph }  /\  E. w  z  =  {
w } ) }
33 19.42v 1858 . . . . . . . . 9  |-  ( E. w ( z  =  { x  |  ph }  /\  z  =  {
w } )  <->  ( z  =  { x  |  ph }  /\  E. w  z  =  { w }
) )
3433bicomi 193 . . . . . . . 8  |-  ( ( z  =  { x  |  ph }  /\  E. w  z  =  {
w } )  <->  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) )
3534abbii 2408 . . . . . . 7  |-  { z  |  ( z  =  { x  |  ph }  /\  E. w  z  =  { w }
) }  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3632, 35eqtri 2316 . . . . . 6  |-  ( { z  |  z  =  { x  |  ph } }  i^i  { z  |  E. w  z  =  { w } } )  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3731, 36eqtri 2316 . . . . 5  |-  ( { { x  |  ph } }  i^i  Singletons )  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3837unieqi 3853 . . . 4  |-  U. ( { { x  |  ph } }  i^i  Singletons )  =  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) }
3928, 38eqtr4i 2319 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  U. ( { { x  |  ph } }  i^i  Singletons )
4039unieqi 3853 . 2  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. U. ( { {
x  |  ph } }  i^i  Singletons )
411, 40eqtri 2316 1  |-  ( iota
x ph )  =  U. U. ( { { x  |  ph } }  i^i  Singletons )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    i^i cin 3164   {csn 3653   U.cuni 3843   iotacio 5233   Singletonscsingles 24453
This theorem is referenced by:  dffv5  24534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-eprel 4321  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466  df-singleton 24474  df-singles 24475
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