Theorem ecopoprer 3248

Description: Assuming that operation F is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation R, specified by the first hypothesis, is an equivalence relation.

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.cl	|- ((x e. S /\ y e. S) -> (xFy) e. S)
ecopopr.ass	|- ((xFy)Fz) = (xF(yFz))
ecopopr.can	|- ((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z))

Assertion

Ref	Expression
ecopoprer	|- Er R

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

Proof of Theorem ecopoprer

Step	Hyp	Ref	Expression
1		ecopopr.1	. . 3 |- 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)))}
2		ecopopr.com	. . 3 |- (xFy) = (yFx)
3		visset 1350	. . 3 |- g e. V
4	1, 2, 3	ecopoprsym 3246	. 2 |- (fRg -> gRf)
5		ecopopr.cl	. . 3 |- ((x e. S /\ y e. S) -> (xFy) e. S)
6		ecopopr.ass	. . 3 |- ((xFy)Fz) = (xF(yFz))
7		ecopopr.can	. . 3 |- ((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z))
8		visset 1350	. . 3 |- h e. V
9	1, 2, 5, 6, 7, 3, 8	ecopoprtrn 3247	. 2 |- ((fRg /\ gRh) -> fRh)
10	4, 9	ster 3207	1 |- Er R

Syntax hints:   -> wi 2   /\ wa 196  E.wex 678   = weq 797   = wceq 1091   e. wcel 1092  <.cop 1810  {copab 2055   X. cxp 2408  (class class class)co 3001  Er wer 3197

This theorem is referenced by:  enqer 3840  enrer 3970

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  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-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003  df-er 3200

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