Theorem opelxp 3204

Description: Ordered pair membership in a cross product.

Hypothesis

Ref	Expression
opelxp.1	|- B e. V

Assertion

Ref	Expression
opelxp	|- (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D))

Proof of Theorem opelxp

Step	Hyp	Ref	Expression
1		opelxp1 3195	. 2 |- (<.A, B>. e. (C X. D) -> A e. C)
2		pm3.26 319	. 2 |- ((A e. C /\ B e. D) -> A e. C)
3		opeq1 2478	. . . 4 |- (z = A -> <.z, B>. = <.A, B>.)
4	3	eleq1d 1532	. . 3 |- (z = A -> (<.z, B>. e. (C X. D) <-> <.A, B>. e. (C X. D)))
5		eleq1 1526	. . . 4 |- (z = A -> (z e. C <-> A e. C))
6	5	anbi1d 615	. . 3 |- (z = A -> ((z e. C /\ B e. D) <-> (A e. C /\ B e. D)))
7		eqcom 1469	. . . . . . . . . . 11 |- (<.z, B>. = <.x, y>. <-> <.x, y>. = <.z, B>.)
8		visset 1804	. . . . . . . . . . . 12 |- x e. V
9		visset 1804	. . . . . . . . . . . 12 |- y e. V
10		opelxp.1	. . . . . . . . . . . 12 |- B e. V
11	8, 9, 10	opth 2777	. . . . . . . . . . 11 |- (<.x, y>. = <.z, B>. <-> (x = z /\ y = B))
12	7, 11	bitr 173	. . . . . . . . . 10 |- (<.z, B>. = <.x, y>. <-> (x = z /\ y = B))
13	12	anbi1i 480	. . . . . . . . 9 |- ((<.z, B>. = <.x, y>. /\ (x e. C /\ y e. D)) <-> ((x = z /\ y = B) /\ (x e. C /\ y e. D)))
14		an4 505	. . . . . . . . 9 |- (((x = z /\ y = B) /\ (x e. C /\ y e. D)) <-> ((x = z /\ x e. C) /\ (y = B /\ y e. D)))
15	13, 14	bitr 173	. . . . . . . 8 |- ((<.z, B>. = <.x, y>. /\ (x e. C /\ y e. D)) <-> ((x = z /\ x e. C) /\ (y = B /\ y e. D)))
16	15	exbii 1047	. . . . . . 7 |- (E.y(<.z, B>. = <.x, y>. /\ (x e. C /\ y e. D)) <-> E.y((x = z /\ x e. C) /\ (y = B /\ y e. D)))
17		19.42v 1303	. . . . . . 7 |- (E.y((x = z /\ x e. C) /\ (y = B /\ y e. D)) <-> ((x = z /\ x e. C) /\ E.y(y = B /\ y e. D)))
18	16, 17	bitr 173	. . . . . 6 |- (E.y(<.z, B>. = <.x, y>. /\ (x e. C /\ y e. D)) <-> ((x = z /\ x e. C) /\ E.y(y = B /\ y e. D)))
19	18	exbii 1047	. . . . 5 |- (E.xE.y(<.z, B>. = <.x, y>. /\ (x e. C /\ y e. D)) <-> E.x((x = z /\ x e. C) /\ E.y(y = B /\ y e. D)))
20		19.41v 1300	. . . . 5 |- (E.x((x = z /\ x e. C) /\ E.y(y = B /\ y e. D)) <-> (E.x(x = z /\ x e. C) /\ E.y(y = B /\ y e. D)))
21	19, 20	bitr 173	. . . 4 |- (E.xE.y(<.z, B>. = <.x, y>. /\ (x e. C /\ y e. D)) <-> (E.x(x = z /\ x e. C) /\ E.y(y = B /\ y e. D)))
22		elxp 3192	. . . 4 |- (<.z, B>. e. (C X. D) <-> E.xE.y(<.z, B>. = <.x, y>. /\ (x e. C /\ y e. D)))
23		df-clel 1465	. . . . 5 |- (z e. C <-> E.x(x = z /\ x e. C))
24		df-clel 1465	. . . . 5 |- (B e. D <-> E.y(y = B /\ y e. D))
25	23, 24	anbi12i 481	. . . 4 |- ((z e. C /\ B e. D) <-> (E.x(x = z /\ x e. C) /\ E.y(y = B /\ y e. D)))
26	21, 22, 25	3bitr4 183	. . 3 |- (<.z, B>. e. (C X. D) <-> (z e. C /\ B e. D))
27	4, 6, 26	vtoclbg 1839	. 2 |- (A e. C -> (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D)))
28	1, 2, 27	pm5.21nii 677	1 |- (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  Vcvv 1802  <.cop 2401   X. cxp 3158

This theorem is referenced by:  brxp 3205  opelxpg 3206  ralxp 3208  opthprc 3211  elxp3 3214  optocl 3225  relsn 3244  ssxp 3246  xpsspw 3247  inxp 3259  opelres 3356  resiexg 3380  dfima2 3389  cnvxp 3450  xpnz 3452  ssrnres 3467  relssdr 3499  opelf 3625  dff2 3802  dff3 3803  resoprab 3994  oprssdm 4027  curry1 4082  df1st2 4110  df2nd2 4111  brecop 4290  brecop2 4291  ecopoprdm 4293  eceqopreq 4297  xpdom2 4422  xpmapenlem4 4479  xpmapenlem5 4480  mapunen 4482  aceq5lem2 4708  ltpiord 4987  opelcn 5220  opelreal 5221  ruclem13 7465  infxpidmlem5 7499  infxpidmlem7 7501  xplmi 7907  bcthlem32 7964  nvvop 8166  stcat 10353

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-opab 2657  df-xp 3174

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