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Theorem idref 5664
Description: TODO: This is the same as issref 4927 (which has a much longer proof). Should we replace issref 4927 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 2140 . . . 4  |-  ( x  e.  A  |->  <. x ,  x >. )  =  ( x  e.  A  |->  <.
x ,  x >. )
21fmpt 5576 . . 3  |-  ( A. x  e.  A  <. x ,  x >.  e.  R  <->  ( x  e.  A  |->  <.
x ,  x >. ) : A --> R )
3 vex 2692 . . . . . 6  |-  x  e. 
_V
43, 3opex 4157 . . . . 5  |-  <. x ,  x >.  e.  _V
54, 1fnmpti 5257 . . . 4  |-  ( x  e.  A  |->  <. x ,  x >. )  Fn  A
6 df-f 5133 . . . 4  |-  ( ( x  e.  A  |->  <.
x ,  x >. ) : A --> R  <->  ( (
x  e.  A  |->  <.
x ,  x >. )  Fn  A  /\  ran  ( x  e.  A  |-> 
<. x ,  x >. ) 
C_  R ) )
75, 6mpbiran 925 . . 3  |-  ( ( x  e.  A  |->  <.
x ,  x >. ) : A --> R  <->  ran  ( x  e.  A  |->  <. x ,  x >. )  C_  R
)
82, 7bitri 183 . 2  |-  ( A. x  e.  A  <. x ,  x >.  e.  R  <->  ran  ( x  e.  A  |-> 
<. x ,  x >. ) 
C_  R )
9 df-br 3936 . . 3  |-  ( x R x  <->  <. x ,  x >.  e.  R
)
109ralbii 2444 . 2  |-  ( A. x  e.  A  x R x  <->  A. x  e.  A  <. x ,  x >.  e.  R )
11 mptresid 4879 . . . 4  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
123fnasrn 5604 . . . 4  |-  ( x  e.  A  |->  x )  =  ran  ( x  e.  A  |->  <. x ,  x >. )
1311, 12eqtr3i 2163 . . 3  |-  (  _I  |`  A )  =  ran  ( x  e.  A  |-> 
<. x ,  x >. )
1413sseq1i 3126 . 2  |-  ( (  _I  |`  A )  C_  R  <->  ran  ( x  e.  A  |->  <. x ,  x >. )  C_  R )
158, 10, 143bitr4ri 212 1  |-  ( (  _I  |`  A )  C_  R  <->  A. x  e.  A  x R x )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1481   A.wral 2417    C_ wss 3074   <.cop 3533   class class class wbr 3935    |-> cmpt 3995    _I cid 4216   ran crn 4546    |` cres 4547    Fn wfn 5124   -->wf 5125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-id 4221  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137
This theorem is referenced by: (None)
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