MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relbrtpos Structured version   Unicode version

Theorem relbrtpos 6958
Description: The transposition swaps arguments of a three-parameter relation. (Contributed by Mario Carneiro, 3-Nov-2015.)
Assertion
Ref Expression
relbrtpos  |-  ( Rel 
F  ->  ( <. A ,  B >.tpos  F C  <->  <. B ,  A >. F C ) )

Proof of Theorem relbrtpos
StepHypRef Expression
1 reltpos 6952 . . . 4  |-  Rel tpos  F
21a1i 11 . . 3  |-  ( Rel 
F  ->  Rel tpos  F )
3 brrelex2 5028 . . 3  |-  ( ( Rel tpos  F  /\  <. A ,  B >.tpos  F C )  ->  C  e.  _V )
42, 3sylan 469 . 2  |-  ( ( Rel  F  /\  <. A ,  B >.tpos  F C )  ->  C  e.  _V )
5 brrelex2 5028 . 2  |-  ( ( Rel  F  /\  <. B ,  A >. F C )  ->  C  e.  _V )
6 brtpos 6956 . 2  |-  ( C  e.  _V  ->  ( <. A ,  B >.tpos  F C  <->  <. B ,  A >. F C ) )
74, 5, 6pm5.21nd 898 1  |-  ( Rel 
F  ->  ( <. A ,  B >.tpos  F C  <->  <. B ,  A >. F C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1823   _Vcvv 3106   <.cop 4022   class class class wbr 4439   Rel wrel 4993  tpos ctpos 6946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-tpos 6947
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator