Theorem ecopoprsym 3246

Description: Assuming the operation F is commutative, show that the relation R, specified by the first hypothesis, is symmetric.

Hypotheses

Ref	Expression
ecopopr.1	|- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}
ecopopr.com	|- (xFy) = (yFx)
ecopopr.2	|- B e. V

Assertion

Ref	Expression
ecopoprsym	|- (ARB -> BRA)

Distinct variable group(s):   x,y,z,w,v,u,F   x,S,y,z,w,v,u

Proof of Theorem ecopoprsym

Step	Hyp	Ref	Expression
1		ecopopr.2	. . . 4 |- B e. V
2		ecopopr.1	. . . . 5 |- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))}
3		opabssxp 2468	. . . . 5 |- {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ (zFu) = (wFv)))} (_ ((S X. S) X. (S X. S))
4	2, 3	eqsstr 1530	. . . 4 |- R (_ ((S X. S) X. (S X. S))
5	1, 4	brel 2459	. . 3 |- (ARB -> (A e. (S X. S) /\ B e. (S X. S)))
6		cleqid 1102	. . . 4 |- (S X. S) = (S X. S)
7		breq1 2065	. . . . 5 |- (<.f, g>. = A -> (<.f, g>.R<.h, t>. <-> AR<.h, t>.))
8		breq2 2066	. . . . 5 |- (<.f, g>. = A -> (<.h, t>.R<.f, g>. <-> <.h, t>.RA))
9	7, 8	bibi12d 477	. . . 4 |- (<.f, g>. = A -> ((<.f, g>.R<.h, t>. <-> <.h, t>.R<.f, g>.) <-> (AR<.h, t>. <-> <.h, t>.RA)))
10		breq2 2066	. . . . 5 |- (<.h, t>. = B -> (AR<.h, t>. <-> ARB))
11		breq1 2065	. . . . 5 |- (<.h, t>. = B -> (<.h, t>.RA <-> BRA))
12	10, 11	bibi12d 477	. . . 4 |- (<.h, t>. = B -> ((AR<.h, t>. <-> <.h, t>.RA) <-> (ARB <-> BRA)))
13	2	ecopopreq 3244	. . . . . 6 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S)) -> (<.f, g>.R<.h, t>. <-> (fFt) = (gFh)))
14		visset 1350	. . . . . . . . 9 |- f e. V
15		visset 1350	. . . . . . . . 9 |- t e. V
16		ecopopr.com	. . . . . . . . 9 |- (xFy) = (yFx)
17	14, 15, 16	caoprcom 3067	. . . . . . . 8 |- (fFt) = (tFf)
18		visset 1350	. . . . . . . . 9 |- g e. V
19		visset 1350	. . . . . . . . 9 |- h e. V
20	18, 19, 16	caoprcom 3067	. . . . . . . 8 |- (gFh) = (hFg)
21	17, 20	cleq12i 1114	. . . . . . 7 |- ((fFt) = (gFh) <-> (tFf) = (hFg))
22		cleqcom 1103	. . . . . . 7 |- ((tFf) = (hFg) <-> (hFg) = (tFf))
23	21, 22	bitr 151	. . . . . 6 |- ((fFt) = (gFh) <-> (hFg) = (tFf))
24	13, 23	syl6bb 414	. . . . 5 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S)) -> (<.f, g>.R<.h, t>. <-> (hFg) = (tFf)))
25	2	ecopopreq 3244	. . . . . 6 |- (((h e. S /\ t e. S) /\ (f e. S /\ g e. S)) -> (<.h, t>.R<.f, g>. <-> (hFg) = (tFf)))
26	25	ancoms 334	. . . . 5 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S)) -> (<.h, t>.R<.f, g>. <-> (hFg) = (tFf)))
27	24, 26	bitr4d 409	. . . 4 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S)) -> (<.f, g>.R<.h, t>. <-> <.h, t>.R<.f, g>.))
28	6, 9, 12, 27	2optocl 2470	. . 3 |- ((A e. (S X. S) /\ B e. (S X. S)) -> (ARB <-> BRA))
29	5, 28	syl 12	. 2 |- (ARB -> (ARB <-> BRA))
30	29	ibi 449	1 |- (ARB -> BRA)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348  <.cop 1810   class class class wbr 2054  {copab 2055   X. cxp 2408  (class class class)co 3001

This theorem is referenced by:  ecopoprer 3248

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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