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

Theorem idsset 30196
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 5070 . 2  |-  Rel  _I
2 relsset 30194 . . 3  |-  Rel  SSet
3 relin1 5059 . . 3  |-  ( Rel 
SSet  ->  Rel  ( SSet  i^i  `' SSet ) )
42, 3ax-mp 5 . 2  |-  Rel  ( SSet  i^i  `' SSet )
5 eqss 3454 . . 3  |-  ( y  =  z  <->  ( y  C_  z  /\  z  C_  y ) )
6 vex 3059 . . . 4  |-  z  e. 
_V
76ideq 5095 . . 3  |-  ( y  _I  z  <->  y  =  z )
8 brin 4441 . . . 4  |-  ( y ( SSet  i^i  `' SSet ) z  <->  ( y SSet z  /\  y `'
SSet z ) )
96brsset 30195 . . . . 5  |-  ( y
SSet z  <->  y  C_  z )
10 vex 3059 . . . . . . 7  |-  y  e. 
_V
1110, 6brcnv 5125 . . . . . 6  |-  ( y `' SSet z  <->  z SSet y )
1210brsset 30195 . . . . . 6  |-  ( z
SSet y  <->  z  C_  y )
1311, 12bitri 249 . . . . 5  |-  ( y `' SSet z  <->  z  C_  y )
149, 13anbi12i 695 . . . 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 5038 1  |-  _I  =  ( SSet  i^i  `' SSet )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1403    i^i cin 3410    C_ wss 3411   class class class wbr 4392    _I cid 4730   `'ccnv 4939   Rel wrel 4945   SSetcsset 30137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-eprel 4731  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fo 5529  df-fv 5531  df-1st 6736  df-2nd 6737  df-txp 30159  df-sset 30161
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator