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Theorem tpossym 6979
Description: Two ways to say a function is symmetric. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
tpossym  |-  ( F  Fn  ( A  X.  A )  ->  (tpos  F  =  F  <->  A. x  e.  A  A. y  e.  A  ( x F y )  =  ( y F x ) ) )
Distinct variable groups:    x, y, A    x, F, y

Proof of Theorem tpossym
StepHypRef Expression
1 tposfn 6976 . . 3  |-  ( F  Fn  ( A  X.  A )  -> tpos  F  Fn  ( A  X.  A
) )
2 eqfnov2 6382 . . 3  |-  ( (tpos 
F  Fn  ( A  X.  A )  /\  F  Fn  ( A  X.  A ) )  -> 
(tpos  F  =  F  <->  A. x  e.  A  A. y  e.  A  (
xtpos  F y )  =  ( x F y ) ) )
31, 2mpancom 667 . 2  |-  ( F  Fn  ( A  X.  A )  ->  (tpos  F  =  F  <->  A. x  e.  A  A. y  e.  A  ( xtpos  F y )  =  ( x F y ) ) )
4 eqcom 2463 . . . 4  |-  ( ( xtpos  F y )  =  ( x F y )  <->  ( x F y )  =  ( xtpos  F y ) )
5 ovtpos 6962 . . . . 5  |-  ( xtpos 
F y )  =  ( y F x )
65eqeq2i 2472 . . . 4  |-  ( ( x F y )  =  ( xtpos  F
y )  <->  ( x F y )  =  ( y F x ) )
74, 6bitri 249 . . 3  |-  ( ( xtpos  F y )  =  ( x F y )  <->  ( x F y )  =  ( y F x ) )
872ralbii 2886 . 2  |-  ( A. x  e.  A  A. y  e.  A  (
xtpos  F y )  =  ( x F y )  <->  A. x  e.  A  A. y  e.  A  ( x F y )  =  ( y F x ) )
93, 8syl6bb 261 1  |-  ( F  Fn  ( A  X.  A )  ->  (tpos  F  =  F  <->  A. x  e.  A  A. y  e.  A  ( x F y )  =  ( y F x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398   A.wral 2804    X. cxp 4986    Fn wfn 5565  (class class class)co 6270  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-csb 3421  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-iun 4317  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-f 5574  df-fo 5576  df-fv 5578  df-ov 6273  df-tpos 6947
This theorem is referenced by:  xmettpos  21018
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