Theorem idref 5929

Description: TODO: This is the same as issref 5145 (which has a much longer proof). Should we replace issref 5145 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 2232	. . . 4 |- ( x e. A |-> <. x , x >. ) = ( x e. A |-> <. x , x >. )
2	1	fmpt 5827	. . 3 |- ( A. x e. A <. x , x >. e. R <-> ( x e. A |-> <. x , x >. ) : A --> R )
3		vex 2816	. . . . . 6 |- x e. _V
4	3, 3	opex 4345	. . . . 5 |- <. x , x >. e. _V
5	4, 1	fnmpti 5487	. . . 4 |- ( x e. A |-> <. x , x >. ) Fn A
6		df-f 5356	. . . 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 949	. . 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 4110	. . 3 |- ( x R x <-> <. x , x >. e. R )
10	9	ralbii 2548	. 2 |- ( A. x e. A x R x <-> A. x e. A <. x , x >. e. R )
11		mptresid 5092	. . . . 5 |- ( _I |` A ) = ( x e. A |-> x )
12	11	eqcomi 2236	. . . 4 |- ( x e. A |-> x ) = ( _I |` A )
13	3	fnasrn 5856	. . . 4 |- ( x e. A |-> x ) = ran ( x e. A |-> <. x , x >. )
14	12, 13	eqtr3i 2255	. . 3 |- ( _I |` A ) = ran ( x e. A |-> <. x , x >. )
15	14	sseq1i 3264	. 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 2203   A.wral 2520   C_ wss 3211   <.cop 3692   class class class wbr 4109   |-> cmpt 4171   _I cid 4409   ran crn 4750   |` cres 4751   Fn wfn 5347   -->wf 5348

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360

This theorem is referenced by: (None)