Theorem ecopopreq 4308

Description: This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation R (specified by the hypothesis) in terms of its operation F.

Hypothesis

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)))}

Assertion

Ref	Expression
ecopopreq	|- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>.R<.C, D>. <-> (AFD) = (BFC)))

Distinct variable groups:   x,y,z,w,v,u,F   x,S,y,z,w,v,u   x,A,y,z,w,v,u   x,B,y,z,w,v,u   x,C,y,z,w,v,u   x,D,y,z,w,v,u

Proof of Theorem ecopopreq

Step	Hyp	Ref	Expression
1		opreq12 3970	. . . 4 |- ((z = A /\ u = D) -> (zFu) = (AFD))
2		opreq12 3970	. . . 4 |- ((w = B /\ v = C) -> (wFv) = (BFC))
3	1, 2	eqeqan12d 1490	. . 3 |- (((z = A /\ u = D) /\ (w = B /\ v = C)) -> ((zFu) = (wFv) <-> (AFD) = (BFC)))
4	3	an42s 509	. 2 |- (((z = A /\ w = B) /\ (v = C /\ u = D)) -> ((zFu) = (wFv) <-> (AFD) = (BFC)))
5		ecopopr.1	. 2 |- 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)))}
6	4, 5	opbrop 3238	1 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>.R<.C, D>. <-> (AFD) = (BFC)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  <.cop 2411   class class class wbr 2619  {copab 2666   X. cxp 3168  (class class class)co 3963

This theorem is referenced by:  ecopoprdm 4309  ecopoprsym 4310  ecopoprtrn 4311  enqbreq 5044  enrbreq 5174

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-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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

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