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Theorem dfiota3 30761
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 5553 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 abeq1 2581 . . . . 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 1843 . . . . . 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 3034 . . . . . . . . 9  |-  y  e. 
_V
5 sneq 3969 . . . . . . . . . 10  |-  ( w  =  y  ->  { w }  =  { y } )
65eqeq2d 2481 . . . . . . . . 9  |-  ( w  =  y  ->  ( { x  |  ph }  =  { w }  <->  { x  |  ph }  =  {
y } ) )
74, 6ceqsexv 3070 . . . . . . . 8  |-  ( E. w ( w  =  y  /\  { x  |  ph }  =  {
w } )  <->  { x  |  ph }  =  {
y } )
8 snex 4641 . . . . . . . . . . 11  |-  { w }  e.  _V
9 eqeq1 2475 . . . . . . . . . . . . 13  |-  ( z  =  { w }  ->  ( z  =  {
x  |  ph }  <->  { w }  =  {
x  |  ph }
) )
10 eleq2 2538 . . . . . . . . . . . . 13  |-  ( z  =  { w }  ->  ( y  e.  z  <-> 
y  e.  { w } ) )
119, 10anbi12d 725 . . . . . . . . . . . 12  |-  ( z  =  { w }  ->  ( ( z  =  { x  |  ph }  /\  y  e.  z )  <->  ( { w }  =  { x  |  ph }  /\  y  e.  { w } ) ) )
12 eqcom 2478 . . . . . . . . . . . . 13  |-  ( { w }  =  {
x  |  ph }  <->  { x  |  ph }  =  { w } )
13 elsn 3973 . . . . . . . . . . . . . 14  |-  ( y  e.  { w }  <->  y  =  w )
14 equcom 1870 . . . . . . . . . . . . . 14  |-  ( y  =  w  <->  w  =  y )
1513, 14bitri 257 . . . . . . . . . . . . 13  |-  ( y  e.  { w }  <->  w  =  y )
1612, 15anbi12ci 712 . . . . . . . . . . . 12  |-  ( ( { w }  =  { x  |  ph }  /\  y  e.  { w } )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) )
1711, 16syl6bb 269 . . . . . . . . . . 11  |-  ( z  =  { w }  ->  ( ( z  =  { x  |  ph }  /\  y  e.  z )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) ) )
188, 17ceqsexv 3070 . . . . . . . . . 10  |-  ( E. z ( z  =  { w }  /\  ( z  =  {
x  |  ph }  /\  y  e.  z
) )  <->  ( w  =  y  /\  { x  |  ph }  =  {
w } ) )
19 an13 816 . . . . . . . . . . 11  |-  ( ( z  =  { w }  /\  ( z  =  { x  |  ph }  /\  y  e.  z ) )  <->  ( y  e.  z  /\  (
z  =  { x  |  ph }  /\  z  =  { w } ) ) )
2019exbii 1726 . . . . . . . . . 10  |-  ( E. z ( z  =  { w }  /\  ( z  =  {
x  |  ph }  /\  y  e.  z
) )  <->  E. z
( y  e.  z  /\  ( z  =  { x  |  ph }  /\  z  =  {
w } ) ) )
2118, 20bitr3i 259 . . . . . . . . 9  |-  ( ( w  =  y  /\  { x  |  ph }  =  { w } )  <->  E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
2221exbii 1726 . . . . . . . 8  |-  ( E. w ( w  =  y  /\  { x  |  ph }  =  {
w } )  <->  E. w E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
237, 22bitr3i 259 . . . . . . 7  |-  ( { x  |  ph }  =  { y }  <->  E. w E. z ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
24 excom 1944 . . . . . . 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 257 . . . . . 6  |-  ( { x  |  ph }  =  { y }  <->  E. z E. w ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
26 eluniab 4201 . . . . . 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 285 . . . . 5  |-  ( { x  |  ph }  =  { y }  <->  y  e.  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) } )
282, 27mpgbir 1681 . . . 4  |-  { y  |  { x  | 
ph }  =  {
y } }  =  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) }
29 df-sn 3960 . . . . . . 7  |-  { {
x  |  ph } }  =  { z  |  z  =  {
x  |  ph } }
30 dfsingles2 30759 . . . . . . 7  |-  Singletons  =  {
z  |  E. w  z  =  { w } }
3129, 30ineq12i 3623 . . . . . 6  |-  ( { { x  |  ph } }  i^i  Singletons )  =  ( { z  |  z  =  { x  | 
ph } }  i^i  { z  |  E. w  z  =  { w } } )
32 inab 3702 . . . . . . 7  |-  ( { z  |  z  =  { x  |  ph } }  i^i  { z  |  E. w  z  =  { w } } )  =  {
z  |  ( z  =  { x  | 
ph }  /\  E. w  z  =  {
w } ) }
33 19.42v 1842 . . . . . . . . 9  |-  ( E. w ( z  =  { x  |  ph }  /\  z  =  {
w } )  <->  ( z  =  { x  |  ph }  /\  E. w  z  =  { w }
) )
3433bicomi 207 . . . . . . . 8  |-  ( ( z  =  { x  |  ph }  /\  E. w  z  =  {
w } )  <->  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) )
3534abbii 2587 . . . . . . 7  |-  { z  |  ( z  =  { x  |  ph }  /\  E. w  z  =  { w }
) }  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3632, 35eqtri 2493 . . . . . 6  |-  ( { z  |  z  =  { x  |  ph } }  i^i  { z  |  E. w  z  =  { w } } )  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3731, 36eqtri 2493 . . . . 5  |-  ( { { x  |  ph } }  i^i  Singletons )  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3837unieqi 4199 . . . 4  |-  U. ( { { x  |  ph } }  i^i  Singletons )  =  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) }
3928, 38eqtr4i 2496 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  U. ( { { x  |  ph } }  i^i  Singletons )
4039unieqi 4199 . 2  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. U. ( { {
x  |  ph } }  i^i  Singletons )
411, 40eqtri 2493 1  |-  ( iota
x ph )  =  U. U. ( { { x  |  ph } }  i^i  Singletons )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457    i^i cin 3389   {csn 3959   U.cuni 4190   iotacio 5551   Singletonscsingles 30676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-symdif 3654  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-eprel 4750  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595  df-fv 5597  df-1st 6812  df-2nd 6813  df-txp 30691  df-singleton 30699  df-singles 30700
This theorem is referenced by:  dffv5  30762
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