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Theorem tpostpos2 6467
Description: Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
tpostpos2  |-  ( ( Rel  F  /\  Rel  dom 
F )  -> tpos tpos  F  =  F )

Proof of Theorem tpostpos2
StepHypRef Expression
1 tpostpos 6466 . 2  |- tpos tpos  F  =  ( F  i^i  (
( ( _V  X.  _V )  u.  { (/) } )  X.  _V )
)
2 relrelss 5360 . . . 4  |-  ( ( Rel  F  /\  Rel  dom 
F )  <->  F  C_  (
( _V  X.  _V )  X.  _V ) )
3 ssun1 3478 . . . . . 6  |-  ( _V 
X.  _V )  C_  (
( _V  X.  _V )  u.  { (/) } )
4 xpss1 4951 . . . . . 6  |-  ( ( _V  X.  _V )  C_  ( ( _V  X.  _V )  u.  { (/) } )  ->  ( ( _V  X.  _V )  X. 
_V )  C_  (
( ( _V  X.  _V )  u.  { (/) } )  X.  _V )
)
53, 4ax-mp 8 . . . . 5  |-  ( ( _V  X.  _V )  X.  _V )  C_  (
( ( _V  X.  _V )  u.  { (/) } )  X.  _V )
6 sstr 3324 . . . . 5  |-  ( ( F  C_  ( ( _V  X.  _V )  X. 
_V )  /\  (
( _V  X.  _V )  X.  _V )  C_  ( ( ( _V 
X.  _V )  u.  { (/)
} )  X.  _V ) )  ->  F  C_  ( ( ( _V 
X.  _V )  u.  { (/)
} )  X.  _V ) )
75, 6mpan2 653 . . . 4  |-  ( F 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  F  C_  (
( ( _V  X.  _V )  u.  { (/) } )  X.  _V )
)
82, 7sylbi 188 . . 3  |-  ( ( Rel  F  /\  Rel  dom 
F )  ->  F  C_  ( ( ( _V 
X.  _V )  u.  { (/)
} )  X.  _V ) )
9 df-ss 3302 . . 3  |-  ( F 
C_  ( ( ( _V  X.  _V )  u.  { (/) } )  X. 
_V )  <->  ( F  i^i  ( ( ( _V 
X.  _V )  u.  { (/)
} )  X.  _V ) )  =  F )
108, 9sylib 189 . 2  |-  ( ( Rel  F  /\  Rel  dom 
F )  ->  ( F  i^i  ( ( ( _V  X.  _V )  u.  { (/) } )  X. 
_V ) )  =  F )
111, 10syl5eq 2456 1  |-  ( ( Rel  F  /\  Rel  dom 
F )  -> tpos tpos  F  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649   _Vcvv 2924    u. cun 3286    i^i cin 3287    C_ wss 3288   (/)c0 3596   {csn 3782    X. cxp 4843   dom cdm 4845   Rel wrel 4850  tpos ctpos 6445
This theorem is referenced by:  2oppchomf  13913
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-fv 5429  df-tpos 6446
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