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Theorem tpostpos2 6974
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 6973 . 2  |- tpos tpos  F  =  ( F  i^i  (
( ( _V  X.  _V )  u.  { (/) } )  X.  _V )
)
2 relrelss 5517 . . . 4  |-  ( ( Rel  F  /\  Rel  dom 
F )  <->  F  C_  (
( _V  X.  _V )  X.  _V ) )
3 ssun1 3649 . . . . . 6  |-  ( _V 
X.  _V )  C_  (
( _V  X.  _V )  u.  { (/) } )
4 xpss1 5097 . . . . . 6  |-  ( ( _V  X.  _V )  C_  ( ( _V  X.  _V )  u.  { (/) } )  ->  ( ( _V  X.  _V )  X. 
_V )  C_  (
( ( _V  X.  _V )  u.  { (/) } )  X.  _V )
)
53, 4ax-mp 5 . . . . 5  |-  ( ( _V  X.  _V )  X.  _V )  C_  (
( ( _V  X.  _V )  u.  { (/) } )  X.  _V )
6 sstr 3494 . . . . 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 671 . . . 4  |-  ( F 
C_  ( ( _V 
X.  _V )  X.  _V )  ->  F  C_  (
( ( _V  X.  _V )  u.  { (/) } )  X.  _V )
)
82, 7sylbi 195 . . 3  |-  ( ( Rel  F  /\  Rel  dom 
F )  ->  F  C_  ( ( ( _V 
X.  _V )  u.  { (/)
} )  X.  _V ) )
9 df-ss 3472 . . 3  |-  ( F 
C_  ( ( ( _V  X.  _V )  u.  { (/) } )  X. 
_V )  <->  ( F  i^i  ( ( ( _V 
X.  _V )  u.  { (/)
} )  X.  _V ) )  =  F )
108, 9sylib 196 . 2  |-  ( ( Rel  F  /\  Rel  dom 
F )  ->  ( F  i^i  ( ( ( _V  X.  _V )  u.  { (/) } )  X. 
_V ) )  =  F )
111, 10syl5eq 2494 1  |-  ( ( Rel  F  /\  Rel  dom 
F )  -> tpos tpos  F  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381   _Vcvv 3093    u. cun 3456    i^i cin 3457    C_ wss 3458   (/)c0 3767   {csn 4010    X. cxp 4983   dom cdm 4985   Rel wrel 4990  tpos ctpos 6952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-fv 5582  df-tpos 6953
This theorem is referenced by:  2oppchomf  14991  mattpostpos  18823
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