ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tpossym Unicode version

Theorem tpossym 6253
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 6250 . . 3  |-  ( F  Fn  ( A  X.  A )  -> tpos  F  Fn  ( A  X.  A
) )
2 eqfnov2 5958 . . 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 420 . 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 2172 . . . 4  |-  ( ( xtpos  F y )  =  ( x F y )  <->  ( x F y )  =  ( xtpos  F y ) )
5 vex 2733 . . . . . 6  |-  x  e. 
_V
6 vex 2733 . . . . . 6  |-  y  e. 
_V
7 ovtposg 6236 . . . . . 6  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( xtpos  F y )  =  ( y F x ) )
85, 6, 7mp2an 424 . . . . 5  |-  ( xtpos 
F y )  =  ( y F x )
98eqeq2i 2181 . . . 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 2478 . 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, 11bitrdi 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 1348    e. wcel 2141   A.wral 2448   _Vcvv 2730    X. cxp 4607    Fn wfn 5191  (class class class)co 5851  tpos ctpos 6221
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fo 5202  df-fv 5204  df-ov 5854  df-tpos 6222
This theorem is referenced by:  xmettpos  13129
  Copyright terms: Public domain W3C validator