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Theorem tpossym 6166
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 6163 . . 3  |-  ( F  Fn  ( A  X.  A )  -> tpos  F  Fn  ( A  X.  A
) )
2 eqfnov2 5871 . . 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 418 . 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 2139 . . . 4  |-  ( ( xtpos  F y )  =  ( x F y )  <->  ( x F y )  =  ( xtpos  F y ) )
5 vex 2684 . . . . . 6  |-  x  e. 
_V
6 vex 2684 . . . . . 6  |-  y  e. 
_V
7 ovtposg 6149 . . . . . 6  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( xtpos  F y )  =  ( y F x ) )
85, 6, 7mp2an 422 . . . . 5  |-  ( xtpos 
F y )  =  ( y F x )
98eqeq2i 2148 . . . 4  |-  ( ( x F y )  =  ( xtpos  F
y )  <->  ( x F y )  =  ( y F x ) )
104, 9bitri 183 . . 3  |-  ( ( xtpos  F y )  =  ( x F y )  <->  ( x F y )  =  ( y F x ) )
11102ralbii 2441 . 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 ) )
123, 11syl6bb 195 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 104    = wceq 1331    e. wcel 1480   A.wral 2414   _Vcvv 2681    X. cxp 4532    Fn wfn 5113  (class class class)co 5767  tpos ctpos 6134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-ov 5770  df-tpos 6135
This theorem is referenced by:  xmettpos  12528
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