Theorem opbrop 3244

Description: Ordered pair membership in a relation. Special case.

Hypotheses

Ref	Expression
opbrop.1	|- (((z = A /\ w = B) /\ (v = C /\ u = D)) -> (ph <-> ps))
opbrop.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>.) /\ ph))}

Assertion

Ref	Expression
opbrop	|- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>.R<.C, D>. <-> ps))

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

Proof of Theorem opbrop

Step	Hyp	Ref	Expression
1		opbrop.1	. . . . 5 |- (((z = A /\ w = B) /\ (v = C /\ u = D)) -> (ph <-> ps))
2	1	copsex4g 2800	. . . 4 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (E.zE.wE.vE.u((<.A, B>. = <.z, w>. /\ <.C, D>. = <.v, u>.) /\ ph) <-> ps))
3	2	anbi2d 618	. . 3 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (((<.A, B>. e. (S X. S) /\ <.C, D>. e. (S X. S)) /\ E.zE.wE.vE.u((<.A, B>. = <.z, w>. /\ <.C, D>. = <.v, u>.) /\ ph)) <-> ((<.A, B>. e. (S X. S) /\ <.C, D>. e. (S X. S)) /\ ps)))
4		opex 2788	. . . 4 |- <.A, B>. e. V
5		opex 2788	. . . 4 |- <.C, D>. e. V
6		eleq1 1537	. . . . . 6 |- (x = <.A, B>. -> (x e. (S X. S) <-> <.A, B>. e. (S X. S)))
7	6	anbi1d 619	. . . . 5 |- (x = <.A, B>. -> ((x e. (S X. S) /\ y e. (S X. S)) <-> (<.A, B>. e. (S X. S) /\ y e. (S X. S))))
8		eqeq1 1484	. . . . . . . 8 |- (x = <.A, B>. -> (x = <.z, w>. <-> <.A, B>. = <.z, w>.))
9	8	anbi1d 619	. . . . . . 7 |- (x = <.A, B>. -> ((x = <.z, w>. /\ y = <.v, u>.) <-> (<.A, B>. = <.z, w>. /\ y = <.v, u>.)))
10	9	anbi1d 619	. . . . . 6 |- (x = <.A, B>. -> (((x = <.z, w>. /\ y = <.v, u>.) /\ ph) <-> ((<.A, B>. = <.z, w>. /\ y = <.v, u>.) /\ ph)))
11	10	4exbidv 1285	. . . . 5 |- (x = <.A, B>. -> (E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ ph) <-> E.zE.wE.vE.u((<.A, B>. = <.z, w>. /\ y = <.v, u>.) /\ ph)))
12	7, 11	anbi12d 630	. . . 4 |- (x = <.A, B>. -> (((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ ph)) <-> ((<.A, B>. e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((<.A, B>. = <.z, w>. /\ y = <.v, u>.) /\ ph))))
13		eleq1 1537	. . . . . 6 |- (y = <.C, D>. -> (y e. (S X. S) <-> <.C, D>. e. (S X. S)))
14	13	anbi2d 618	. . . . 5 |- (y = <.C, D>. -> ((<.A, B>. e. (S X. S) /\ y e. (S X. S)) <-> (<.A, B>. e. (S X. S) /\ <.C, D>. e. (S X. S))))
15		eqeq1 1484	. . . . . . . 8 |- (y = <.C, D>. -> (y = <.v, u>. <-> <.C, D>. = <.v, u>.))
16	15	anbi2d 618	. . . . . . 7 |- (y = <.C, D>. -> ((<.A, B>. = <.z, w>. /\ y = <.v, u>.) <-> (<.A, B>. = <.z, w>. /\ <.C, D>. = <.v, u>.)))
17	16	anbi1d 619	. . . . . 6 |- (y = <.C, D>. -> (((<.A, B>. = <.z, w>. /\ y = <.v, u>.) /\ ph) <-> ((<.A, B>. = <.z, w>. /\ <.C, D>. = <.v, u>.) /\ ph)))
18	17	4exbidv 1285	. . . . 5 |- (y = <.C, D>. -> (E.zE.wE.vE.u((<.A, B>. = <.z, w>. /\ y = <.v, u>.) /\ ph) <-> E.zE.wE.vE.u((<.A, B>. = <.z, w>. /\ <.C, D>. = <.v, u>.) /\ ph)))
19	14, 18	anbi12d 630	. . . 4 |- (y = <.C, D>. -> (((<.A, B>. e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((<.A, B>. = <.z, w>. /\ y = <.v, u>.) /\ ph)) <-> ((<.A, B>. e. (S X. S) /\ <.C, D>. e. (S X. S)) /\ E.zE.wE.vE.u((<.A, B>. = <.z, w>. /\ <.C, D>. = <.v, u>.) /\ ph))))
20		opbrop.2	. . . 4 |- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u((x = <.z, w>. /\ y = <.v, u>.) /\ ph))}
21	4, 5, 12, 19, 20	brab 2827	. . 3 |- (<.A, B>.R<.C, D>. <-> ((<.A, B>. e. (S X. S) /\ <.C, D>. e. (S X. S)) /\ E.zE.wE.vE.u((<.A, B>. = <.z, w>. /\ <.C, D>. = <.v, u>.) /\ ph)))
22	3, 21	syl5bb 534	. 2 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>.R<.C, D>. <-> ((<.A, B>. e. (S X. S) /\ <.C, D>. e. (S X. S)) /\ ps)))
23		opelxpi 3223	. . . 4 |- ((A e. S /\ B e. S) -> <.A, B>. e. (S X. S))
24		opelxpi 3223	. . . 4 |- ((C e. S /\ D e. S) -> <.C, D>. e. (S X. S))
25	23, 24	anim12i 333	. . 3 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>. e. (S X. S) /\ <.C, D>. e. (S X. S)))
26	25	biantrurd 729	. 2 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (ps <-> ((<.A, B>. e. (S X. S) /\ <.C, D>. e. (S X. S)) /\ ps)))
27	22, 26	bitr4d 533	1 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>.R<.C, D>. <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  <.cop 2415   class class class wbr 2624  {copab 2671   X. cxp 3174

This theorem is referenced by:  ecopopreq 4314  oprec 4324

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190

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