Theorem ecopoprer 4312

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 groups:   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 1813	. . 3 |- g e. V
4	1, 2, 3	ecopoprsym 4310	. 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 1813	. . 3 |- h e. V
9	1, 2, 5, 6, 7, 3, 8	ecopoprtrn 4311	. 2 |- ((fRg /\ gRh) -> fRh)
10	4, 9	ster 4268	1 |- Er R

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  <.cop 2411  {copab 2666   X. cxp 3168  (class class class)co 3963  Er wer 4258

This theorem is referenced by:  enqer 5046  enrer 5176

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965  df-er 4261

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