MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfid3 Unicode version

Theorem dfid3 4441
Description: A stronger version of df-id 4440 that doesn't require  x and  y to be distinct. Ordinarily, we wouldn't use this as a definition, since non-distinct dummy variables would make soundness verification more difficult (as the proof here shows). The proof can be instructive in showing how distinct variable requirements may be eliminated, a task that is not necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
dfid3  |-  _I  =  { <. x ,  y
>.  |  x  =  y }

Proof of Theorem dfid3
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-id 4440 . 2  |-  _I  =  { <. x ,  z
>.  |  x  =  z }
2 ancom 438 . . . . . . . . . . 11  |-  ( ( w  =  <. x ,  z >.  /\  x  =  z )  <->  ( x  =  z  /\  w  =  <. x ,  z
>. ) )
3 equcom 1687 . . . . . . . . . . . 12  |-  ( x  =  z  <->  z  =  x )
43anbi1i 677 . . . . . . . . . . 11  |-  ( ( x  =  z  /\  w  =  <. x ,  z >. )  <->  ( z  =  x  /\  w  =  <. x ,  z
>. ) )
52, 4bitri 241 . . . . . . . . . 10  |-  ( ( w  =  <. x ,  z >.  /\  x  =  z )  <->  ( z  =  x  /\  w  =  <. x ,  z
>. ) )
65exbii 1589 . . . . . . . . 9  |-  ( E. z ( w  = 
<. x ,  z >.  /\  x  =  z
)  <->  E. z ( z  =  x  /\  w  =  <. x ,  z
>. ) )
7 vex 2903 . . . . . . . . . 10  |-  x  e. 
_V
8 opeq2 3928 . . . . . . . . . . 11  |-  ( z  =  x  ->  <. x ,  z >.  =  <. x ,  x >. )
98eqeq2d 2399 . . . . . . . . . 10  |-  ( z  =  x  ->  (
w  =  <. x ,  z >.  <->  w  =  <. x ,  x >. ) )
107, 9ceqsexv 2935 . . . . . . . . 9  |-  ( E. z ( z  =  x  /\  w  = 
<. x ,  z >.
)  <->  w  =  <. x ,  x >. )
11 equid 1683 . . . . . . . . . 10  |-  x  =  x
1211biantru 492 . . . . . . . . 9  |-  ( w  =  <. x ,  x >.  <-> 
( w  =  <. x ,  x >.  /\  x  =  x ) )
136, 10, 123bitri 263 . . . . . . . 8  |-  ( E. z ( w  = 
<. x ,  z >.  /\  x  =  z
)  <->  ( w  = 
<. x ,  x >.  /\  x  =  x ) )
1413exbii 1589 . . . . . . 7  |-  ( E. x E. z ( w  =  <. x ,  z >.  /\  x  =  z )  <->  E. x
( w  =  <. x ,  x >.  /\  x  =  x ) )
15 nfe1 1739 . . . . . . . 8  |-  F/ x E. x ( w  = 
<. x ,  x >.  /\  x  =  x )
161519.9 1791 . . . . . . 7  |-  ( E. x E. x ( w  =  <. x ,  x >.  /\  x  =  x )  <->  E. x
( w  =  <. x ,  x >.  /\  x  =  x ) )
1714, 16bitr4i 244 . . . . . 6  |-  ( E. x E. z ( w  =  <. x ,  z >.  /\  x  =  z )  <->  E. x E. x ( w  = 
<. x ,  x >.  /\  x  =  x ) )
18 opeq2 3928 . . . . . . . . . . 11  |-  ( x  =  y  ->  <. x ,  x >.  =  <. x ,  y >. )
1918eqeq2d 2399 . . . . . . . . . 10  |-  ( x  =  y  ->  (
w  =  <. x ,  x >.  <->  w  =  <. x ,  y >. )
)
20 equequ2 1693 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x  =  x  <->  x  =  y ) )
2119, 20anbi12d 692 . . . . . . . . 9  |-  ( x  =  y  ->  (
( w  =  <. x ,  x >.  /\  x  =  x )  <->  ( w  =  <. x ,  y
>.  /\  x  =  y ) ) )
2221sps 1762 . . . . . . . 8  |-  ( A. x  x  =  y  ->  ( ( w  = 
<. x ,  x >.  /\  x  =  x )  <-> 
( w  =  <. x ,  y >.  /\  x  =  y ) ) )
2322drex1 2008 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( E. x ( w  =  <. x ,  x >.  /\  x  =  x )  <->  E. y
( w  =  <. x ,  y >.  /\  x  =  y ) ) )
2423drex2 2009 . . . . . 6  |-  ( A. x  x  =  y  ->  ( E. x E. x ( w  = 
<. x ,  x >.  /\  x  =  x )  <->  E. x E. y ( w  =  <. x ,  y >.  /\  x  =  y ) ) )
2517, 24syl5bb 249 . . . . 5  |-  ( A. x  x  =  y  ->  ( E. x E. z ( w  = 
<. x ,  z >.  /\  x  =  z
)  <->  E. x E. y
( w  =  <. x ,  y >.  /\  x  =  y ) ) )
26 nfnae 2002 . . . . . 6  |-  F/ x  -.  A. x  x  =  y
27 nfnae 2002 . . . . . . 7  |-  F/ y  -.  A. x  x  =  y
28 nfcvd 2525 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  F/_ y w )
29 nfcvf2 2547 . . . . . . . . . 10  |-  ( -. 
A. x  x  =  y  ->  F/_ y x )
30 nfcvd 2525 . . . . . . . . . 10  |-  ( -. 
A. x  x  =  y  ->  F/_ y z )
3129, 30nfopd 3944 . . . . . . . . 9  |-  ( -. 
A. x  x  =  y  ->  F/_ y <.
x ,  z >.
)
3228, 31nfeqd 2538 . . . . . . . 8  |-  ( -. 
A. x  x  =  y  ->  F/ y  w  =  <. x ,  z >. )
3329, 30nfeqd 2538 . . . . . . . 8  |-  ( -. 
A. x  x  =  y  ->  F/ y  x  =  z )
3432, 33nfand 1833 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  F/ y
( w  =  <. x ,  z >.  /\  x  =  z ) )
35 opeq2 3928 . . . . . . . . . 10  |-  ( z  =  y  ->  <. x ,  z >.  =  <. x ,  y >. )
3635eqeq2d 2399 . . . . . . . . 9  |-  ( z  =  y  ->  (
w  =  <. x ,  z >.  <->  w  =  <. x ,  y >.
) )
37 equequ2 1693 . . . . . . . . 9  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
3836, 37anbi12d 692 . . . . . . . 8  |-  ( z  =  y  ->  (
( w  =  <. x ,  z >.  /\  x  =  z )  <->  ( w  =  <. x ,  y
>.  /\  x  =  y ) ) )
3938a1i 11 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  ( z  =  y  ->  ( ( w  =  <. x ,  z >.  /\  x  =  z )  <->  ( w  =  <. x ,  y
>.  /\  x  =  y ) ) ) )
4027, 34, 39cbvexd 2044 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  ( E. z ( w  = 
<. x ,  z >.  /\  x  =  z
)  <->  E. y ( w  =  <. x ,  y
>.  /\  x  =  y ) ) )
4126, 40exbid 1781 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( E. x E. z ( w  =  <. x ,  z
>.  /\  x  =  z )  <->  E. x E. y
( w  =  <. x ,  y >.  /\  x  =  y ) ) )
4225, 41pm2.61i 158 . . . 4  |-  ( E. x E. z ( w  =  <. x ,  z >.  /\  x  =  z )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  x  =  y
) )
4342abbii 2500 . . 3  |-  { w  |  E. x E. z
( w  =  <. x ,  z >.  /\  x  =  z ) }  =  { w  |  E. x E. y
( w  =  <. x ,  y >.  /\  x  =  y ) }
44 df-opab 4209 . . 3  |-  { <. x ,  z >.  |  x  =  z }  =  { w  |  E. x E. z ( w  =  <. x ,  z
>.  /\  x  =  z ) }
45 df-opab 4209 . . 3  |-  { <. x ,  y >.  |  x  =  y }  =  { w  |  E. x E. y ( w  =  <. x ,  y
>.  /\  x  =  y ) }
4643, 44, 453eqtr4i 2418 . 2  |-  { <. x ,  z >.  |  x  =  z }  =  { <. x ,  y
>.  |  x  =  y }
471, 46eqtri 2408 1  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649   {cab 2374   <.cop 3761   {copab 4207    _I cid 4435
This theorem is referenced by:  dfid2  4442  reli  4943  opabresid  5135  ider  6876  cnmptid  17615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-opab 4209  df-id 4440
  Copyright terms: Public domain W3C validator