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Theorem coi1 5204
Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
Assertion
Ref Expression
coi1  |-  ( Rel 
A  ->  ( A  o.  _I  )  =  A )

Proof of Theorem coi1
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5187 . 2  |-  Rel  ( A  o.  _I  )
2 vex 2804 . . . . . 6  |-  x  e. 
_V
3 vex 2804 . . . . . 6  |-  y  e. 
_V
42, 3opelco 4869 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<->  E. z ( x  _I  z  /\  z A y ) )
5 vex 2804 . . . . . . . . . 10  |-  z  e. 
_V
65ideq 4852 . . . . . . . . 9  |-  ( x  _I  z  <->  x  =  z )
7 equcom 1665 . . . . . . . . 9  |-  ( x  =  z  <->  z  =  x )
86, 7bitri 240 . . . . . . . 8  |-  ( x  _I  z  <->  z  =  x )
98anbi1i 676 . . . . . . 7  |-  ( ( x  _I  z  /\  z A y )  <->  ( z  =  x  /\  z A y ) )
109exbii 1572 . . . . . 6  |-  ( E. z ( x  _I  z  /\  z A y )  <->  E. z
( z  =  x  /\  z A y ) )
11 breq1 4042 . . . . . . 7  |-  ( z  =  x  ->  (
z A y  <->  x A
y ) )
122, 11ceqsexv 2836 . . . . . 6  |-  ( E. z ( z  =  x  /\  z A y )  <->  x A
y )
1310, 12bitri 240 . . . . 5  |-  ( E. z ( x  _I  z  /\  z A y )  <->  x A
y )
144, 13bitri 240 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<->  x A y )
15 df-br 4040 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
1614, 15bitri 240 . . 3  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<-> 
<. x ,  y >.  e.  A )
1716eqrelriv 4796 . 2  |-  ( ( Rel  ( A  o.  _I  )  /\  Rel  A
)  ->  ( A  o.  _I  )  =  A )
181, 17mpan 651 1  |-  ( Rel 
A  ->  ( A  o.  _I  )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039    _I cid 4320    o. ccom 4709   Rel wrel 4710
This theorem is referenced by:  coi2  5205  coires1  5206  relcoi1  5217  fcoi1  5431  cocnv  26496  mvdco  27491
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-rab 2565  df-v 2803  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-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-co 4714
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