Theorem tpossym 6441

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

Step	Hyp	Ref	Expression
1		tposfn 6438	. . 3 |- ( F Fn ( A X. A ) -> tpos F Fn ( A X. A ) )
2		eqfnov2 6128	. . 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 ) ) )
3	1, 2	mpancom 422	. 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 2233	. . . 4 |- ( ( xtpos F y ) = ( x F y ) <-> ( x F y ) = ( xtpos F y ) )
5		vex 2805	. . . . . 6 |- x e. _V
6		vex 2805	. . . . . 6 |- y e. _V
7		ovtposg 6424	. . . . . 6 |- ( ( x e. _V /\ y e. _V ) -> ( xtpos F y ) = ( y F x ) )
8	5, 6, 7	mp2an 426	. . . . 5 |- ( xtpos F y ) = ( y F x )
9	8	eqeq2i 2242	. . . 4 |- ( ( x F y ) = ( xtpos F y ) <-> ( x F y ) = ( y F x ) )
10	4, 9	bitri 184	. . 3 |- ( ( xtpos F y ) = ( x F y ) <-> ( x F y ) = ( y F x ) )
11	10	2ralbii 2540	. 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 ) )
12	3, 11	bitrdi 196	1 |- ( F Fn ( A X. A ) -> (tpos F = F <-> A. x e. A A. y e. A ( x F y ) = ( y F x ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   X. cxp 4723   Fn wfn 5321  (class class class)co 6017  tpos ctpos 6409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-ov 6020  df-tpos 6410

This theorem is referenced by:  xmettpos  15093