Theorem idref 6499

Description: TODO: This is the same as issref 5509 (which has a much longer proof). Should we replace issref 5509 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 2622	. . . 4 |- ( x e. A |-> <. x , x >. ) = ( x e. A |-> <. x , x >. )
2	1	fmpt 6381	. . 3 |- ( A. x e. A <. x , x >. e. R <-> ( x e. A |-> <. x , x >. ) : A --> R )
3		opex 4932	. . . . 5 |- <. x , x >. e. _V
4	3, 1	fnmpti 6022	. . . 4 |- ( x e. A |-> <. x , x >. ) Fn A
5		df-f 5892	. . . 4 |- ( ( x e. A |-> <. x , x >. ) : A --> R <-> ( ( x e. A |-> <. x , x >. ) Fn A /\ ran ( x e. A |-> <. x , x >. ) C_ R ) )
6	4, 5	mpbiran 953	. . 3 |- ( ( x e. A |-> <. x , x >. ) : A --> R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
7	2, 6	bitri 264	. 2 |- ( A. x e. A <. x , x >. e. R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
8		df-br 4654	. . 3 |- ( x R x <-> <. x , x >. e. R )
9	8	ralbii 2980	. 2 |- ( A. x e. A x R x <-> A. x e. A <. x , x >. e. R )
10		mptresid 5456	. . . 4 |- ( x e. A |-> x ) = ( _I |` A )
11		vex 3203	. . . . 5 |- x e. _V
12	11	fnasrn 6411	. . . 4 |- ( x e. A |-> x ) = ran ( x e. A |-> <. x , x >. )
13	10, 12	eqtr3i 2646	. . 3 |- ( _I |` A ) = ran ( x e. A |-> <. x , x >. )
14	13	sseq1i 3629	. 2 |- ( ( _I |` A ) C_ R <-> ran ( x e. A |-> <. x , x >. ) C_ R )
15	7, 9, 14	3bitr4ri 293	1 |- ( ( _I |` A ) C_ R <-> A. x e. A x R x )

Syntax hints:   <-> wb 196   e. wcel 1990   A.wral 2912   C_ wss 3574   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  retos  19964  filnetlem2  32374