Theorem idref 5803

Description: TODO: This is the same as issref 5052 (which has a much longer proof). Should we replace issref 5052 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 2196	. . . 4 |- ( x e. A |-> <. x , x >. ) = ( x e. A |-> <. x , x >. )
2	1	fmpt 5712	. . 3 |- ( A. x e. A <. x , x >. e. R <-> ( x e. A |-> <. x , x >. ) : A --> R )
3		vex 2766	. . . . . 6 |- x e. _V
4	3, 3	opex 4262	. . . . 5 |- <. x , x >. e. _V
5	4, 1	fnmpti 5386	. . . 4 |- ( x e. A |-> <. x , x >. ) Fn A
6		df-f 5262	. . . 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 942	. . 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 4034	. . 3 |- ( x R x <-> <. x , x >. e. R )
10	9	ralbii 2503	. 2 |- ( A. x e. A x R x <-> A. x e. A <. x , x >. e. R )
11		mptresid 5000	. . . . 5 |- ( _I |` A ) = ( x e. A |-> x )
12	11	eqcomi 2200	. . . 4 |- ( x e. A |-> x ) = ( _I |` A )
13	3	fnasrn 5740	. . . 4 |- ( x e. A |-> x ) = ran ( x e. A |-> <. x , x >. )
14	12, 13	eqtr3i 2219	. . 3 |- ( _I |` A ) = ran ( x e. A |-> <. x , x >. )
15	14	sseq1i 3209	. 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 2167   A.wral 2475   C_ wss 3157   <.cop 3625   class class class wbr 4033   |-> cmpt 4094   _I cid 4323   ran crn 4664   |` cres 4665   Fn wfn 5253   -->wf 5254

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266

This theorem is referenced by: (None)