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Theorem idref 6170
Description: TODO: This is the same as issref 5231 (which has a much longer proof). Should we replace issref 5231 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
StepHypRef Expression
1 eqid 2461 . . . 4  |-  ( x  e.  A  |->  <. x ,  x >. )  =  ( x  e.  A  |->  <.
x ,  x >. )
21fmpt 6065 . . 3  |-  ( A. x  e.  A  <. x ,  x >.  e.  R  <->  ( x  e.  A  |->  <.
x ,  x >. ) : A --> R )
3 opex 4677 . . . . 5  |-  <. x ,  x >.  e.  _V
43, 1fnmpti 5727 . . . 4  |-  ( x  e.  A  |->  <. x ,  x >. )  Fn  A
5 df-f 5604 . . . 4  |-  ( ( x  e.  A  |->  <.
x ,  x >. ) : A --> R  <->  ( (
x  e.  A  |->  <.
x ,  x >. )  Fn  A  /\  ran  ( x  e.  A  |-> 
<. x ,  x >. ) 
C_  R ) )
64, 5mpbiran 934 . . 3  |-  ( ( x  e.  A  |->  <.
x ,  x >. ) : A --> R  <->  ran  ( x  e.  A  |->  <. x ,  x >. )  C_  R
)
72, 6bitri 257 . 2  |-  ( A. x  e.  A  <. x ,  x >.  e.  R  <->  ran  ( x  e.  A  |-> 
<. x ,  x >. ) 
C_  R )
8 df-br 4416 . . 3  |-  ( x R x  <->  <. x ,  x >.  e.  R
)
98ralbii 2830 . 2  |-  ( A. x  e.  A  x R x  <->  A. x  e.  A  <. x ,  x >.  e.  R )
10 mptresid 5177 . . . 4  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
11 vex 3059 . . . . 5  |-  x  e. 
_V
1211fnasrn 6093 . . . 4  |-  ( x  e.  A  |->  x )  =  ran  ( x  e.  A  |->  <. x ,  x >. )
1310, 12eqtr3i 2485 . . 3  |-  (  _I  |`  A )  =  ran  ( x  e.  A  |-> 
<. x ,  x >. )
1413sseq1i 3467 . 2  |-  ( (  _I  |`  A )  C_  R  <->  ran  ( x  e.  A  |->  <. x ,  x >. )  C_  R )
157, 9, 143bitr4ri 286 1  |-  ( (  _I  |`  A )  C_  R  <->  A. x  e.  A  x R x )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    e. wcel 1897   A.wral 2748    C_ wss 3415   <.cop 3985   class class class wbr 4415    |-> cmpt 4474    _I cid 4762   ran crn 4853    |` cres 4854    Fn wfn 5595   -->wf 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608
This theorem is referenced by:  retos  19234  filnetlem2  31083
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