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Theorem idref 5775
Description: TODO: This is the same as issref 5072 (which has a much longer proof). Should we replace issref 5072 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 2296 . . . 4  |-  ( x  e.  A  |->  <. x ,  x >. )  =  ( x  e.  A  |->  <.
x ,  x >. )
21fmpt 5697 . . 3  |-  ( A. x  e.  A  <. x ,  x >.  e.  R  <->  ( x  e.  A  |->  <.
x ,  x >. ) : A --> R )
3 opex 4253 . . . . 5  |-  <. x ,  x >.  e.  _V
43, 1fnmpti 5388 . . . 4  |-  ( x  e.  A  |->  <. x ,  x >. )  Fn  A
5 df-f 5275 . . . 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 4040 . . 3  |-  ( x R x  <->  <. x ,  x >.  e.  R
)
98ralbii 2580 . 2  |-  ( A. x  e.  A  x R x  <->  A. x  e.  A  <. x ,  x >.  e.  R )
10 mptresid 5020 . . . 4  |-  ( x  e.  A  |->  x )  =  (  _I  |`  A )
11 vex 2804 . . . . 5  |-  x  e. 
_V
1211fnasrn 5718 . . . 4  |-  ( x  e.  A  |->  x )  =  ran  ( x  e.  A  |->  <. x ,  x >. )
1310, 12eqtr3i 2318 . . 3  |-  (  _I  |`  A )  =  ran  ( x  e.  A  |-> 
<. x ,  x >. )
1413sseq1i 3215 . 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 1696   A.wral 2556    C_ wss 3165   <.cop 3656   class class class wbr 4039    e. cmpt 4093    _I cid 4320   ran crn 4706    |` cres 4707    Fn wfn 5266   -->wf 5267
This theorem is referenced by:  preoref22  25332  preoran2  25333  dfps2  25392  filnetlem2  26431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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