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

Theorem idsset 25735
Description:  _I is equal to  SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
Assertion
Ref Expression
idsset  |-  _I  =  ( SSet  i^i  `' SSet )

Proof of Theorem idsset
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reli 5002 . 2  |-  Rel  _I
2 relsset 25733 . . 3  |-  Rel  SSet
3 relin1 4992 . . 3  |-  ( Rel 
SSet  ->  Rel  ( SSet  i^i  `' SSet ) )
42, 3ax-mp 8 . 2  |-  Rel  ( SSet  i^i  `' SSet )
5 eqss 3363 . . 3  |-  ( y  =  z  <->  ( y  C_  z  /\  z  C_  y ) )
6 vex 2959 . . . 4  |-  z  e. 
_V
76ideq 5025 . . 3  |-  ( y  _I  z  <->  y  =  z )
8 brin 4259 . . . 4  |-  ( y ( SSet  i^i  `' SSet ) z  <->  ( y SSet z  /\  y `'
SSet z ) )
96brsset 25734 . . . . 5  |-  ( y
SSet z  <->  y  C_  z )
10 vex 2959 . . . . . . 7  |-  y  e. 
_V
1110, 6brcnv 5055 . . . . . 6  |-  ( y `' SSet z  <->  z SSet y )
1210brsset 25734 . . . . . 6  |-  ( z
SSet y  <->  z  C_  y )
1311, 12bitri 241 . . . . 5  |-  ( y `' SSet z  <->  z  C_  y )
149, 13anbi12i 679 . . . 4  |-  ( ( y SSet z  /\  y `' SSet z )  <->  ( y  C_  z  /\  z  C_  y ) )
158, 14bitri 241 . . 3  |-  ( y ( SSet  i^i  `' SSet ) z  <->  ( y  C_  z  /\  z  C_  y ) )
165, 7, 153bitr4i 269 . 2  |-  ( y  _I  z  <->  y ( SSet  i^i  `' SSet )
z )
171, 4, 16eqbrriv 4971 1  |-  _I  =  ( SSet  i^i  `' SSet )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    i^i cin 3319    C_ wss 3320   class class class wbr 4212    _I cid 4493   `'ccnv 4877   Rel wrel 4883   SSetcsset 25676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-eprel 4494  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-txp 25698  df-sset 25700
  Copyright terms: Public domain W3C validator