Theorem ecopoprsym 4294

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 groups:   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 3224	. . . . 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 2081	. . . 4 |- R (_ ((S X. S) X. (S X. S))
5	1, 4	brel 3213	. . 3 |- (ARB -> (A e. (S X. S) /\ B e. (S X. S)))
6		eqid 1468	. . . 4 |- (S X. S) = (S X. S)
7		breq1 2612	. . . . 5 |- (<.f, g>. = A -> (<.f, g>.R<.h, t>. <-> AR<.h, t>.))
8		breq2 2613	. . . . 5 |- (<.f, g>. = A -> (<.h, t>.R<.f, g>. <-> <.h, t>.RA))
9	7, 8	bibi12d 627	. . . 4 |- (<.f, g>. = A -> ((<.f, g>.R<.h, t>. <-> <.h, t>.R<.f, g>.) <-> (AR<.h, t>. <-> <.h, t>.RA)))
10		breq2 2613	. . . . 5 |- (<.h, t>. = B -> (AR<.h, t>. <-> ARB))
11		breq1 2612	. . . . 5 |- (<.h, t>. = B -> (<.h, t>.RA <-> BRA))
12	10, 11	bibi12d 627	. . . 4 |- (<.h, t>. = B -> ((AR<.h, t>. <-> <.h, t>.RA) <-> (ARB <-> BRA)))
13	2	ecopopreq 4292	. . . . . 6 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S)) -> (<.f, g>.R<.h, t>. <-> (fFt) = (gFh)))
14		visset 1804	. . . . . . . . 9 |- f e. V
15		visset 1804	. . . . . . . . 9 |- t e. V
16		ecopopr.com	. . . . . . . . 9 |- (xFy) = (yFx)
17	14, 15, 16	caoprcom 4039	. . . . . . . 8 |- (fFt) = (tFf)
18		visset 1804	. . . . . . . . 9 |- g e. V
19		visset 1804	. . . . . . . . 9 |- h e. V
20	18, 19, 16	caoprcom 4039	. . . . . . . 8 |- (gFh) = (hFg)
21	17, 20	eqeq12i 1480	. . . . . . 7 |- ((fFt) = (gFh) <-> (tFf) = (hFg))
22		eqcom 1469	. . . . . . 7 |- ((tFf) = (hFg) <-> (hFg) = (tFf))
23	21, 22	bitr 173	. . . . . 6 |- ((fFt) = (gFh) <-> (hFg) = (tFf))
24	13, 23	syl6bb 534	. . . . 5 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S)) -> (<.f, g>.R<.h, t>. <-> (hFg) = (tFf)))
25	2	ecopopreq 4292	. . . . . 6 |- (((h e. S /\ t e. S) /\ (f e. S /\ g e. S)) -> (<.h, t>.R<.f, g>. <-> (hFg) = (tFf)))
26	25	ancoms 436	. . . . 5 |- (((f e. S /\ g e. S) /\ (h e. S /\ t e. S)) -> (<.h, t>.R<.f, g>. <-> (hFg) = (tFf)))
27	24, 26	bitr4d 529	. . . 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 3226	. . 3 |- ((A e. (S X. S) /\ B e. (S X. S)) -> (ARB <-> BRA))
29	5, 28	syl 10	. 2 |- (ARB -> (ARB <-> BRA))
30	29	ibi 590	1 |- (ARB -> BRA)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802  <.cop 2401   class class class wbr 2609  {copab 2656   X. cxp 3158  (class class class)co 3948

This theorem is referenced by:  ecopoprer 4296

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fv 3188  df-opr 3950

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