Theorem dirref 17235

Description: A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Hypothesis

Ref	Expression
dirref.1	|- X = dom R

Assertion

Ref	Expression
dirref	|- ( ( R e. DirRel /\ A e. X ) -> A R A )

Proof of Theorem dirref

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- A = A
2		resieq 5407	. . . . 5 |- ( ( A e. X /\ A e. X ) -> ( A ( _I |` X ) A <-> A = A ) )
3	2	anidms 677	. . . 4 |- ( A e. X -> ( A ( _I |` X ) A <-> A = A ) )
4	1, 3	mpbiri 248	. . 3 |- ( A e. X -> A ( _I |` X ) A )
5		dirref.1	. . . . . . 7 |- X = dom R
6		dirdm 17234	. . . . . . 7 |- ( R e. DirRel -> dom R = U. U. R )
7	5, 6	syl5eq 2668	. . . . . 6 |- ( R e. DirRel -> X = U. U. R )
8	7	reseq2d 5396	. . . . 5 |- ( R e. DirRel -> ( _I |` X ) = ( _I |` U. U. R ) )
9		eqid 2622	. . . . . . . . 9 |- U. U. R = U. U. R
10	9	isdir 17232	. . . . . . . 8 |- ( R e. DirRel -> ( R e. DirRel <-> ( ( Rel R /\ ( _I |` U. U. R ) C_ R ) /\ ( ( R o. R ) C_ R /\ ( U. U. R X. U. U. R ) C_ ( `' R o. R ) ) ) ) )
11	10	ibi 256	. . . . . . 7 |- ( R e. DirRel -> ( ( Rel R /\ ( _I |` U. U. R ) C_ R ) /\ ( ( R o. R ) C_ R /\ ( U. U. R X. U. U. R ) C_ ( `' R o. R ) ) ) )
12	11	simpld 475	. . . . . 6 |- ( R e. DirRel -> ( Rel R /\ ( _I |` U. U. R ) C_ R ) )
13	12	simprd 479	. . . . 5 |- ( R e. DirRel -> ( _I |` U. U. R ) C_ R )
14	8, 13	eqsstrd 3639	. . . 4 |- ( R e. DirRel -> ( _I |` X ) C_ R )
15	14	ssbrd 4696	. . 3 |- ( R e. DirRel -> ( A ( _I |` X ) A -> A R A ) )
16	4, 15	syl5 34	. 2 |- ( R e. DirRel -> ( A e. X -> A R A ) )
17	16	imp 445	1 |- ( ( R e. DirRel /\ A e. X ) -> A R A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   U.cuni 4436   class class class wbr 4653   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   |` 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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-dir 17230

This theorem is referenced by:  tailini  32371