Theorem idref 5890

Description: TODO: This is the same as issref 5115 (which has a much longer proof). Should we replace issref 5115 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

Assertion

Ref	Expression
idref	|- ( ( _I |` A ) C_ R <-> A. x e. A x R x )

Distinct variable groups:   x, A   x, R

Proof of Theorem idref

Step	Hyp	Ref	Expression
1		eqid 2229	. . . 4 |- ( x e. A |-> <. x , x >. ) = ( x e. A |-> <. x , x >. )
2	1	fmpt 5791	. . 3 |- ( A. x e. A <. x , x >. e. R <-> ( x e. A |-> <. x , x >. ) : A --> R )
3		vex 2803	. . . . . 6 |- x e. _V
4	3, 3	opex 4317	. . . . 5 |- <. x , x >. e. _V
5	4, 1	fnmpti 5456	. . . 4 |- ( x e. A |-> <. x , x >. ) Fn A
6		df-f 5326	. . . 4 |- ( ( x e. A |-> <. x , x >. ) : A --> R <-> ( ( x e. A |-> <. x , x >. ) Fn A /\ ran ( x e. A |-> <. x , x >. ) C_ R ) )
7	5, 6	mpbiran 946	. . 3 |- ( ( x e. A |-> <. x , x >. ) : A --> R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
8	2, 7	bitri 184	. 2 |- ( A. x e. A <. x , x >. e. R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
9		df-br 4085	. . 3 |- ( x R x <-> <. x , x >. e. R )
10	9	ralbii 2536	. 2 |- ( A. x e. A x R x <-> A. x e. A <. x , x >. e. R )
11		mptresid 5063	. . . . 5 |- ( _I |` A ) = ( x e. A |-> x )
12	11	eqcomi 2233	. . . 4 |- ( x e. A |-> x ) = ( _I |` A )
13	3	fnasrn 5819	. . . 4 |- ( x e. A |-> x ) = ran ( x e. A |-> <. x , x >. )
14	12, 13	eqtr3i 2252	. . 3 |- ( _I |` A ) = ran ( x e. A |-> <. x , x >. )
15	14	sseq1i 3251	. 2 |- ( ( _I |` A ) C_ R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
16	8, 10, 15	3bitr4ri 213	1 |- ( ( _I |` A ) C_ R <-> A. x e. A x R x )

Syntax hints:   <-> wb 105   e. wcel 2200   A.wral 2508   C_ wss 3198   <.cop 3670   class class class wbr 4084   |-> cmpt 4146   _I cid 4381   ran crn 4722   |` cres 4723   Fn wfn 5317   -->wf 5318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4203  ax-pow 4260  ax-pr 4295

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330

This theorem is referenced by: (None)