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Theorem idref 5518
Description: TODO: This is the same as issref 4801 (which has a much longer proof). Should we replace issref 4801 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 2088 . . . 4  |-  ( x  e.  A  |->  <. x ,  x >. )  =  ( x  e.  A  |->  <.
x ,  x >. )
21fmpt 5433 . . 3  |-  ( A. x  e.  A  <. x ,  x >.  e.  R  <->  ( x  e.  A  |->  <.
x ,  x >. ) : A --> R )
3 vex 2622 . . . . . 6  |-  x  e. 
_V
43, 3opex 4047 . . . . 5  |-  <. x ,  x >.  e.  _V
54, 1fnmpti 5128 . . . 4  |-  ( x  e.  A  |->  <. x ,  x >. )  Fn  A
6 df-f 5006 . . . 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 886 . . 3  |-  ( ( x  e.  A  |->  <.
x ,  x >. ) : A --> R  <->  ran  ( x  e.  A  |->  <. x ,  x >. )  C_  R
)
82, 7bitri 182 . 2  |-  ( A. x  e.  A  <. x ,  x >.  e.  R  <->  ran  ( x  e.  A  |-> 
<. x ,  x >. ) 
C_  R )
9 df-br 3838 . . 3  |-  ( x R x  <->  <. x ,  x >.  e.  R
)
109ralbii 2384 . 2  |-  ( A. x  e.  A  x R x  <->  A. x  e.  A  <. x ,  x >.  e.  R )
11 mptresid 4753 . . . 4  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
123fnasrn 5459 . . . 4  |-  ( x  e.  A  |->  x )  =  ran  ( x  e.  A  |->  <. x ,  x >. )
1311, 12eqtr3i 2110 . . 3  |-  (  _I  |`  A )  =  ran  ( x  e.  A  |-> 
<. x ,  x >. )
1413sseq1i 3048 . 2  |-  ( (  _I  |`  A )  C_  R  <->  ran  ( x  e.  A  |->  <. x ,  x >. )  C_  R )
158, 10, 143bitr4ri 211 1  |-  ( (  _I  |`  A )  C_  R  <->  A. x  e.  A  x R x )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1438   A.wral 2359    C_ wss 2997   <.cop 3444   class class class wbr 3837    |-> cmpt 3891    _I cid 4106   ran crn 4429    |` cres 4430    Fn wfn 4997   -->wf 4998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010
This theorem is referenced by: (None)
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