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Theorem tpossym 6266
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 6263 . . 3  |-  ( F  Fn  ( A  X.  A )  -> tpos  F  Fn  ( A  X.  A
) )
2 eqfnov2 5951 . . 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 650 . 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 2285 . . . 4  |-  ( ( xtpos  F y )  =  ( x F y )  <->  ( x F y )  =  ( xtpos  F y ) )
5 ovtpos 6249 . . . . 5  |-  ( xtpos 
F y )  =  ( y F x )
65eqeq2i 2293 . . . 4  |-  ( ( x F y )  =  ( xtpos  F
y )  <->  ( x F y )  =  ( y F x ) )
74, 6bitri 240 . . 3  |-  ( ( xtpos  F y )  =  ( x F y )  <->  ( x F y )  =  ( y F x ) )
872ralbii 2569 . 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 252 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 176    = wceq 1623   A.wral 2543    X. cxp 4687    Fn wfn 5250  (class class class)co 5858  tpos ctpos 6233
This theorem is referenced by:  xmettpos  17913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-tpos 6234
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