Theorem idref 5658

Description: TODO: This is the same as issref 4921 (which has a much longer proof). Should we replace issref 4921 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 2139	. . . 4 |- ( x e. A |-> <. x , x >. ) = ( x e. A |-> <. x , x >. )
2	1	fmpt 5570	. . 3 |- ( A. x e. A <. x , x >. e. R <-> ( x e. A |-> <. x , x >. ) : A --> R )
3		vex 2689	. . . . . 6 |- x e. _V
4	3, 3	opex 4151	. . . . 5 |- <. x , x >. e. _V
5	4, 1	fnmpti 5251	. . . 4 |- ( x e. A |-> <. x , x >. ) Fn A
6		df-f 5127	. . . 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 924	. . 3 |- ( ( x e. A |-> <. x , x >. ) : A --> R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
8	2, 7	bitri 183	. 2 |- ( A. x e. A <. x , x >. e. R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
9		df-br 3930	. . 3 |- ( x R x <-> <. x , x >. e. R )
10	9	ralbii 2441	. 2 |- ( A. x e. A x R x <-> A. x e. A <. x , x >. e. R )
11		mptresid 4873	. . . 4 |- ( x e. A |-> x ) = ( _I |` A )
12	3	fnasrn 5598	. . . 4 |- ( x e. A |-> x ) = ran ( x e. A |-> <. x , x >. )
13	11, 12	eqtr3i 2162	. . 3 |- ( _I |` A ) = ran ( x e. A |-> <. x , x >. )
14	13	sseq1i 3123	. 2 |- ( ( _I |` A ) C_ R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
15	8, 10, 14	3bitr4ri 212	1 |- ( ( _I |` A ) C_ R <-> A. x e. A x R x )

Syntax hints:   <-> wb 104   e. wcel 1480   A.wral 2416   C_ wss 3071   <.cop 3530   class class class wbr 3929   |-> cmpt 3989   _I cid 4210   ran crn 4540   |` cres 4541   Fn wfn 5118   -->wf 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131

This theorem is referenced by: (None)