Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfiota3 Structured version   Unicode version

Theorem dfiota3 29426
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 5551 . 2  |-  ( iota
x ph )  =  U. { y  |  {
x  |  ph }  =  { y } }
2 abeq1 2592 . . . . 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 1950 . . . . . 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 3116 . . . . . . . . 9  |-  y  e. 
_V
5 sneq 4037 . . . . . . . . . 10  |-  ( w  =  y  ->  { w }  =  { y } )
65eqeq2d 2481 . . . . . . . . 9  |-  ( w  =  y  ->  ( { x  |  ph }  =  { w }  <->  { x  |  ph }  =  {
y } ) )
74, 6ceqsexv 3150 . . . . . . . 8  |-  ( E. w ( w  =  y  /\  { x  |  ph }  =  {
w } )  <->  { x  |  ph }  =  {
y } )
8 snex 4688 . . . . . . . . . . 11  |-  { w }  e.  _V
9 eqeq1 2471 . . . . . . . . . . . . 13  |-  ( z  =  { w }  ->  ( z  =  {
x  |  ph }  <->  { w }  =  {
x  |  ph }
) )
10 eleq2 2540 . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . 13  |-  ( { w }  =  {
x  |  ph }  <->  { x  |  ph }  =  { w } )
13 elsn 4041 . . . . . . . . . . . . . 14  |-  ( y  e.  { w }  <->  y  =  w )
14 equcom 1743 . . . . . . . . . . . . . 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 3150 . . . . . . . . . 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 1644 . . . . . . . . . 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 1644 . . . . . . . 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 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 249 . . . . . 6  |-  ( { x  |  ph }  =  { y }  <->  E. z E. w ( y  e.  z  /\  ( z  =  { x  | 
ph }  /\  z  =  { w } ) ) )
26 eluniab 4256 . . . . . 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 1605 . . . 4  |-  { y  |  { x  | 
ph }  =  {
y } }  =  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) }
29 df-sn 4028 . . . . . . 7  |-  { {
x  |  ph } }  =  { z  |  z  =  {
x  |  ph } }
30 dfsingles2 29424 . . . . . . 7  |-  Singletons  =  {
z  |  E. w  z  =  { w } }
3129, 30ineq12i 3698 . . . . . 6  |-  ( { { x  |  ph } }  i^i  Singletons )  =  ( { z  |  z  =  { x  | 
ph } }  i^i  { z  |  E. w  z  =  { w } } )
32 inab 3766 . . . . . . 7  |-  ( { z  |  z  =  { x  |  ph } }  i^i  { z  |  E. w  z  =  { w } } )  =  {
z  |  ( z  =  { x  | 
ph }  /\  E. w  z  =  {
w } ) }
33 19.42v 1949 . . . . . . . . 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 2601 . . . . . . 7  |-  { z  |  ( z  =  { x  |  ph }  /\  E. w  z  =  { w }
) }  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3632, 35eqtri 2496 . . . . . 6  |-  ( { z  |  z  =  { x  |  ph } }  i^i  { z  |  E. w  z  =  { w } } )  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3731, 36eqtri 2496 . . . . 5  |-  ( { { x  |  ph } }  i^i  Singletons )  =  {
z  |  E. w
( z  =  {
x  |  ph }  /\  z  =  {
w } ) }
3837unieqi 4254 . . . 4  |-  U. ( { { x  |  ph } }  i^i  Singletons )  =  U. { z  |  E. w ( z  =  { x  |  ph }  /\  z  =  {
w } ) }
3928, 38eqtr4i 2499 . . 3  |-  { y  |  { x  | 
ph }  =  {
y } }  =  U. ( { { x  |  ph } }  i^i  Singletons )
4039unieqi 4254 . 2  |-  U. {
y  |  { x  |  ph }  =  {
y } }  =  U. U. ( { {
x  |  ph } }  i^i  Singletons )
411, 40eqtri 2496 1  |-  ( iota
x ph )  =  U. U. ( { { x  |  ph } }  i^i  Singletons )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    i^i cin 3475   {csn 4027   U.cuni 4245   iotacio 5549   Singletonscsingles 29341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-eprel 4791  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-1st 6785  df-2nd 6786  df-symdif 29321  df-txp 29356  df-singleton 29364  df-singles 29365
This theorem is referenced by:  dffv5  29427
  Copyright terms: Public domain W3C validator