Theorem isdir 17232

Description: A condition for a relation to be a direction. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Hypothesis

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

Assertion

Ref	Expression
isdir	|- ( R e. V -> ( R e. DirRel <-> ( ( Rel R /\ ( _I |` A ) C_ R ) /\ ( ( R o. R ) C_ R /\ ( A X. A ) C_ ( `' R o. R ) ) ) ) )

Proof of Theorem isdir

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		releq 5201	. . . 4 |- ( r = R -> ( Rel r <-> Rel R ) )
2		unieq 4444	. . . . . . . 8 |- ( r = R -> U. r = U. R )
3	2	unieqd 4446	. . . . . . 7 |- ( r = R -> U. U. r = U. U. R )
4		isdir.1	. . . . . . 7 |- A = U. U. R
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( r = R -> U. U. r = A )
6	5	reseq2d 5396	. . . . 5 |- ( r = R -> ( _I |` U. U. r ) = ( _I |` A ) )
7		id 22	. . . . 5 |- ( r = R -> r = R )
8	6, 7	sseq12d 3634	. . . 4 |- ( r = R -> ( ( _I |` U. U. r ) C_ r <-> ( _I |` A ) C_ R ) )
9	1, 8	anbi12d 747	. . 3 |- ( r = R -> ( ( Rel r /\ ( _I |` U. U. r ) C_ r ) <-> ( Rel R /\ ( _I |` A ) C_ R ) ) )
10	7, 7	coeq12d 5286	. . . . 5 |- ( r = R -> ( r o. r ) = ( R o. R ) )
11	10, 7	sseq12d 3634	. . . 4 |- ( r = R -> ( ( r o. r ) C_ r <-> ( R o. R ) C_ R ) )
12	5	sqxpeqd 5141	. . . . 5 |- ( r = R -> ( U. U. r X. U. U. r ) = ( A X. A ) )
13		cnveq 5296	. . . . . 6 |- ( r = R -> `' r = `' R )
14	13, 7	coeq12d 5286	. . . . 5 |- ( r = R -> ( `' r o. r ) = ( `' R o. R ) )
15	12, 14	sseq12d 3634	. . . 4 |- ( r = R -> ( ( U. U. r X. U. U. r ) C_ ( `' r o. r ) <-> ( A X. A ) C_ ( `' R o. R ) ) )
16	11, 15	anbi12d 747	. . 3 |- ( r = R -> ( ( ( r o. r ) C_ r /\ ( U. U. r X. U. U. r ) C_ ( `' r o. r ) ) <-> ( ( R o. R ) C_ R /\ ( A X. A ) C_ ( `' R o. R ) ) ) )
17	9, 16	anbi12d 747	. 2 |- ( r = R -> ( ( ( Rel r /\ ( _I |` U. U. r ) C_ r ) /\ ( ( r o. r ) C_ r /\ ( U. U. r X. U. U. r ) C_ ( `' r o. r ) ) ) <-> ( ( Rel R /\ ( _I |` A ) C_ R ) /\ ( ( R o. R ) C_ R /\ ( A X. A ) C_ ( `' R o. R ) ) ) ) )
18		df-dir 17230	. 2 |- DirRel = { r | ( ( Rel r /\ ( _I |` U. U. r ) C_ r ) /\ ( ( r o. r ) C_ r /\ ( U. U. r X. U. U. r ) C_ ( `' r o. r ) ) ) }
19	17, 18	elab2g 3353	1 |- ( R e. V -> ( R e. DirRel <-> ( ( Rel R /\ ( _I |` A ) C_ R ) /\ ( ( R o. R ) C_ R /\ ( A X. A ) C_ ( `' R o. R ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   U.cuni 4436   _I cid 5023   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   Rel wrel 5119   DirRelcdir 17228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-res 5126  df-dir 17230

This theorem is referenced by:  reldir  17233  dirdm  17234  dirref  17235  dirtr  17236  dirge  17237  tsrdir  17238  filnetlem3  32375