Theorem isdir 10352

Description: A condition for a relation to be a direction. (Contributed by Jeff Hankins, 25-Nov-2009.) )

Hypothesis

Ref	Expression
isdir.1	|- A = U.U.D

Assertion

Ref	Expression
isdir	|- (D e. B -> (D e. Dir <-> ((Rel D /\ ( _I |` A) C_ D) /\ ((D o. D) C_ D /\ A.x e. A A.y e. A E.z e. A (xDz /\ yDz)))))

Distinct variable groups:   x,y,z,A   x,D,y,z

Proof of Theorem isdir

Step	Hyp	Ref	Expression
1		releq 4071	. . . 4 |- (d = D -> (Rel d <-> Rel D))
2		unieq 3185	. . . . . . . 8 |- (d = D -> U.d = U.D)
3	2	unieqd 3188	. . . . . . 7 |- (d = D -> U.U.d = U.U.D)
4		isdir.1	. . . . . . 7 |- A = U.U.D
5	3, 4	syl6eqr 1946	. . . . . 6 |- (d = D -> U.U.d = A)
6		reseq2 4219	. . . . . 6 |- (U.U.d = A -> ( _I |` U.U.d) = ( _I |` A))
7	5, 6	syl 12	. . . . 5 |- (d = D -> ( _I |` U.U.d) = ( _I |` A))
8		id 73	. . . . 5 |- (d = D -> d = D)
9	7, 8	sseq12d 2646	. . . 4 |- (d = D -> (( _I |` U.U.d) C_ d <-> ( _I |` A) C_ D))
10	1, 9	anbi12d 690	. . 3 |- (d = D -> ((Rel d /\ ( _I |` U.U.d) C_ d) <-> (Rel D /\ ( _I |` A) C_ D)))
11		coeq1 4123	. . . . . 6 |- (d = D -> (d o. d) = (D o. d))
12		coeq2 4124	. . . . . 6 |- (d = D -> (D o. d) = (D o. D))
13	11, 12	eqtrd 1925	. . . . 5 |- (d = D -> (d o. d) = (D o. D))
14	13, 8	sseq12d 2646	. . . 4 |- (d = D -> ((d o. d) C_ d <-> (D o. D) C_ D))
15		breq 3340	. . . . . . . 8 |- (d = D -> (xdz <-> xDz))
16		breq 3340	. . . . . . . 8 |- (d = D -> (ydz <-> yDz))
17	15, 16	anbi12d 690	. . . . . . 7 |- (d = D -> ((xdz /\ ydz) <-> (xDz /\ yDz)))
18	5, 17	rexeqbidv 2275	. . . . . 6 |- (d = D -> (E.z e. U.U.d(xdz /\ ydz) <-> E.z e. A (xDz /\ yDz)))
19	5, 18	raleqbidv 2274	. . . . 5 |- (d = D -> (A.y e. U.U.dE.z e. U.U.d(xdz /\ ydz) <-> A.y e. A E.z e. A (xDz /\ yDz)))
20	5, 19	raleqbidv 2274	. . . 4 |- (d = D -> (A.x e. U.U.dA.y e. U.U.dE.z e. U.U.d(xdz /\ ydz) <-> A.x e. A A.y e. A E.z e. A (xDz /\ yDz)))
21	14, 20	anbi12d 690	. . 3 |- (d = D -> (((d o. d) C_ d /\ A.x e. U.U.dA.y e. U.U.dE.z e. U.U.d(xdz /\ ydz)) <-> ((D o. D) C_ D /\ A.x e. A A.y e. A E.z e. A (xDz /\ yDz))))
22	10, 21	anbi12d 690	. 2 |- (d = D -> (((Rel d /\ ( _I |` U.U.d) C_ d) /\ ((d o. d) C_ d /\ A.x e. U.U.dA.y e. U.U.dE.z e. U.U.d(xdz /\ ydz))) <-> ((Rel D /\ ( _I |` A) C_ D) /\ ((D o. D) C_ D /\ A.x e. A A.y e. A E.z e. A (xDz /\ yDz)))))
23		df-dir 10350	. 2 |- Dir = {d | ((Rel d /\ ( _I |` U.U.d) C_ d) /\ ((d o. d) C_ d /\ A.x e. U.U.dA.y e. U.U.dE.z e. U.U.d(xdz /\ ydz)))}
24	22, 23	elab2g 2406	1 |- (D e. B -> (D e. Dir <-> ((Rel D /\ ( _I |` A) C_ D) /\ ((D o. D) C_ D /\ A.x e. A A.y e. A E.z e. A (xDz /\ yDz)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593  U.cuni 3177   class class class wbr 3338   _I cid 3582   |` cres 3988   o. ccom 3990  Rel wrel 3991  Dircdir 10348

This theorem is referenced by:  reldir 10353  dirdm 10354  dirref 10355  dirtr 10356  dirge 10357  tosdir 10358  filnetlem4 15643

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-in 2603  df-ss 2605  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-co 4003  df-res 4006  df-dir 10350

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