Theorem opbrop 2472

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 group(s):   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 1904	. . . 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 468	. . 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 1893	. . . 4 |- <.A, B>. e. V
5		opex 1893	. . . 4 |- <.C, D>. e. V
6		eleq1 1149	. . . . . 6 |- (x = <.A, B>. -> (x e. (S X. S) <-> <.A, B>. e. (S X. S)))
7	6	anbi1d 469	. . . . 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		cleq1 1107	. . . . . . . 8 |- (x = <.A, B>. -> (x = <.z, w>. <-> <.A, B>. = <.z, w>.))
9	8	anbi1d 469	. . . . . . 7 |- (x = <.A, B>. -> ((x = <.z, w>. /\ y = <.v, u>.) <-> (<.A, B>. = <.z, w>. /\ y = <.v, u>.)))
10	9	anbi1d 469	. . . . . 6 |- (x = <.A, B>. -> (((x = <.z, w>. /\ y = <.v, u>.) /\ ph) <-> ((<.A, B>. = <.z, w>. /\ y = <.v, u>.) /\ ph)))
11	10	bi4exdv 940	. . . . 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 476	. . . 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 1149	. . . . . 6 |- (y = <.C, D>. -> (y e. (S X. S) <-> <.C, D>. e. (S X. S)))
14	13	anbi2d 468	. . . . 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		cleq1 1107	. . . . . . . 8 |- (y = <.C, D>. -> (y = <.v, u>. <-> <.C, D>. = <.v, u>.))
16	15	anbi2d 468	. . . . . . 7 |- (y = <.C, D>. -> ((<.A, B>. = <.z, w>. /\ y = <.v, u>.) <-> (<.A, B>. = <.z, w>. /\ <.C, D>. = <.v, u>.)))
17	16	anbi1d 469	. . . . . 6 |- (y = <.C, D>. -> (((<.A, B>. = <.z, w>. /\ y = <.v, u>.) /\ ph) <-> ((<.A, B>. = <.z, w>. /\ <.C, D>. = <.v, u>.) /\ ph)))
18	17	bi4exdv 940	. . . . 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 476	. . . 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 2118	. . 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 410	. 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 2455	. . . 4 |- ((A e. S /\ B e. S) -> <.A, B>. e. (S X. S))
24		opelxpi 2455	. . . 4 |- ((C e. S /\ D e. S) -> <.C, D>. e. (S X. S))
25	23, 24	anim12i 268	. . 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 546	. 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 409	1 |- (((A e. S /\ B e. S) /\ (C e. S /\ D e. S)) -> (<.A, B>.R<.C, D>. <-> ps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  <.cop 1810   class class class wbr 2054  {copab 2055   X. cxp 2408

This theorem is referenced by:  ecopopreq 3244  oprec 3254

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  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-br 2063  df-opab 2098  df-xp 2424

metamath.org