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

Theorem idsset 29105
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 5123 . 2  |-  Rel  _I
2 relsset 29103 . . 3  |-  Rel  SSet
3 relin1 5113 . . 3  |-  ( Rel 
SSet  ->  Rel  ( SSet  i^i  `' SSet ) )
42, 3ax-mp 5 . 2  |-  Rel  ( SSet  i^i  `' SSet )
5 eqss 3514 . . 3  |-  ( y  =  z  <->  ( y  C_  z  /\  z  C_  y ) )
6 vex 3111 . . . 4  |-  z  e. 
_V
76ideq 5148 . . 3  |-  ( y  _I  z  <->  y  =  z )
8 brin 4491 . . . 4  |-  ( y ( SSet  i^i  `' SSet ) z  <->  ( y SSet z  /\  y `'
SSet z ) )
96brsset 29104 . . . . 5  |-  ( y
SSet z  <->  y  C_  z )
10 vex 3111 . . . . . . 7  |-  y  e. 
_V
1110, 6brcnv 5178 . . . . . 6  |-  ( y `' SSet z  <->  z SSet y )
1210brsset 29104 . . . . . 6  |-  ( z
SSet y  <->  z  C_  y )
1311, 12bitri 249 . . . . 5  |-  ( y `' SSet z  <->  z  C_  y )
149, 13anbi12i 697 . . . 4  |-  ( ( y SSet z  /\  y `' SSet z )  <->  ( y  C_  z  /\  z  C_  y ) )
158, 14bitri 249 . . 3  |-  ( y ( SSet  i^i  `' SSet ) z  <->  ( y  C_  z  /\  z  C_  y ) )
165, 7, 153bitr4i 277 . 2  |-  ( y  _I  z  <->  y ( SSet  i^i  `' SSet )
z )
171, 4, 16eqbrriv 5091 1  |-  _I  =  ( SSet  i^i  `' SSet )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1374    i^i cin 3470    C_ wss 3471   class class class wbr 4442    _I cid 4785   `'ccnv 4993   Rel wrel 4999   SSetcsset 29046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-eprel 4786  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-1st 6776  df-2nd 6777  df-txp 29068  df-sset 29070
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator