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Theorem tpossym 6478
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 6475 . . 3  |-  ( F  Fn  ( A  X.  A )  -> tpos  F  Fn  ( A  X.  A
) )
2 eqfnov2 6144 . . 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 651 . 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 2414 . . . 4  |-  ( ( xtpos  F y )  =  ( x F y )  <->  ( x F y )  =  ( xtpos  F y ) )
5 ovtpos 6461 . . . . 5  |-  ( xtpos 
F y )  =  ( y F x )
65eqeq2i 2422 . . . 4  |-  ( ( x F y )  =  ( xtpos  F
y )  <->  ( x F y )  =  ( y F x ) )
74, 6bitri 241 . . 3  |-  ( ( xtpos  F y )  =  ( x F y )  <->  ( x F y )  =  ( y F x ) )
872ralbii 2700 . 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 253 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 set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   A.wral 2674    X. cxp 4843    Fn wfn 5416  (class class class)co 6048  tpos ctpos 6445
This theorem is referenced by:  xmettpos  18340
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-csb 3220  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-iun 4063  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-f 5425  df-fo 5427  df-fv 5429  df-ov 6051  df-tpos 6446
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