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Theorem idsset 24430
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 4813 . 2  |-  Rel  _I
2 relsset 24428 . . 3  |-  Rel  SSet
3 relin1 4803 . . 3  |-  ( Rel 
SSet  ->  Rel  ( SSet  i^i  `' SSet ) )
42, 3ax-mp 8 . 2  |-  Rel  ( SSet  i^i  `' SSet )
5 eqss 3194 . . 3  |-  ( y  =  z  <->  ( y  C_  z  /\  z  C_  y ) )
6 vex 2791 . . . 4  |-  z  e. 
_V
76ideq 4836 . . 3  |-  ( y  _I  z  <->  y  =  z )
8 brin 4070 . . . 4  |-  ( y ( SSet  i^i  `' SSet ) z  <->  ( y SSet z  /\  y `'
SSet z ) )
96brsset 24429 . . . . 5  |-  ( y
SSet z  <->  y  C_  z )
10 vex 2791 . . . . . . 7  |-  y  e. 
_V
1110, 6brcnv 4864 . . . . . 6  |-  ( y `' SSet z  <->  z SSet y )
1210brsset 24429 . . . . . 6  |-  ( z
SSet y  <->  z  C_  y )
1311, 12bitri 240 . . . . 5  |-  ( y `' SSet z  <->  z  C_  y )
149, 13anbi12i 678 . . . 4  |-  ( ( y SSet z  /\  y `' SSet z )  <->  ( y  C_  z  /\  z  C_  y ) )
158, 14bitri 240 . . 3  |-  ( y ( SSet  i^i  `' SSet ) z  <->  ( y  C_  z  /\  z  C_  y ) )
165, 7, 153bitr4i 268 . 2  |-  ( y  _I  z  <->  y ( SSet  i^i  `' SSet )
z )
171, 4, 16eqbrriv 4782 1  |-  _I  =  ( SSet  i^i  `' SSet )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    i^i cin 3151    C_ wss 3152   class class class wbr 4023    _I cid 4304   `'ccnv 4688   Rel wrel 4694   SSetcsset 24375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-txp 24395  df-sset 24397
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