Theorem idref 5417

Description: TODO: This is the same as issref 4727 (which has a much longer proof). Should we replace issref 4727 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 2081	. . . 4 |- ( x e. A |-> <. x , x >. ) = ( x e. A |-> <. x , x >. )
2	1	fmpt 5340	. . 3 |- ( A. x e. A <. x , x >. e. R <-> ( x e. A |-> <. x , x >. ) : A --> R )
3		vex 2604	. . . . . 6 |- x e. _V
4	3, 3	opex 3984	. . . . 5 |- <. x , x >. e. _V
5	4, 1	fnmpti 5047	. . . 4 |- ( x e. A |-> <. x , x >. ) Fn A
6		df-f 4926	. . . 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 881	. . 3 |- ( ( x e. A |-> <. x , x >. ) : A --> R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
8	2, 7	bitri 182	. 2 |- ( A. x e. A <. x , x >. e. R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
9		df-br 3786	. . 3 |- ( x R x <-> <. x , x >. e. R )
10	9	ralbii 2372	. 2 |- ( A. x e. A x R x <-> A. x e. A <. x , x >. e. R )
11		mptresid 4680	. . . 4 |- ( x e. A |-> x ) = ( _I |` A )
12	3	fnasrn 5362	. . . 4 |- ( x e. A |-> x ) = ran ( x e. A |-> <. x , x >. )
13	11, 12	eqtr3i 2103	. . 3 |- ( _I |` A ) = ran ( x e. A |-> <. x , x >. )
14	13	sseq1i 3023	. 2 |- ( ( _I |` A ) C_ R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
15	8, 10, 14	3bitr4ri 211	1 |- ( ( _I |` A ) C_ R <-> A. x e. A x R x )

Syntax hints:   <-> wb 103   e. wcel 1433   A.wral 2348   C_ wss 2973   <.cop 3401   class class class wbr 3785   |-> cmpt 3839   _I cid 4043   ran crn 4364   |` cres 4365   Fn wfn 4917   -->wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930

This theorem is referenced by: (None)