MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  idref Unicode version

Theorem idref 5759
Description: TODO: This is the same as issref 5056 (which has a much longer proof). Should we replace issref 5056 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 2283 . . . 4  |-  ( x  e.  A  |->  <. x ,  x >. )  =  ( x  e.  A  |->  <.
x ,  x >. )
21fmpt 5681 . . 3  |-  ( A. x  e.  A  <. x ,  x >.  e.  R  <->  ( x  e.  A  |->  <.
x ,  x >. ) : A --> R )
3 opex 4237 . . . . 5  |-  <. x ,  x >.  e.  _V
43, 1fnmpti 5372 . . . 4  |-  ( x  e.  A  |->  <. x ,  x >. )  Fn  A
5 df-f 5259 . . . 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 884 . . 3  |-  ( ( x  e.  A  |->  <.
x ,  x >. ) : A --> R  <->  ran  ( x  e.  A  |->  <. x ,  x >. )  C_  R
)
72, 6bitri 240 . 2  |-  ( A. x  e.  A  <. x ,  x >.  e.  R  <->  ran  ( x  e.  A  |-> 
<. x ,  x >. ) 
C_  R )
8 df-br 4024 . . 3  |-  ( x R x  <->  <. x ,  x >.  e.  R
)
98ralbii 2567 . 2  |-  ( A. x  e.  A  x R x  <->  A. x  e.  A  <. x ,  x >.  e.  R )
10 mptresid 5004 . . . 4  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
11 vex 2791 . . . . 5  |-  x  e. 
_V
1211fnasrn 5702 . . . 4  |-  ( x  e.  A  |->  x )  =  ran  ( x  e.  A  |->  <. x ,  x >. )
1310, 12eqtr3i 2305 . . 3  |-  (  _I  |`  A )  =  ran  ( x  e.  A  |-> 
<. x ,  x >. )
1413sseq1i 3202 . 2  |-  ( (  _I  |`  A )  C_  R  <->  ran  ( x  e.  A  |->  <. x ,  x >. )  C_  R )
157, 9, 143bitr4ri 269 1  |-  ( (  _I  |`  A )  C_  R  <->  A. x  e.  A  x R x )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684   A.wral 2543    C_ wss 3152   <.cop 3643   class class class wbr 4023    e. cmpt 4077    _I cid 4304   ran crn 4690    |` cres 4691    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  preoref22  25229  preoran2  25230  dfps2  25289  filnetlem2  26328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
  Copyright terms: Public domain W3C validator