MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coi1 Unicode version

Theorem coi1 5186
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 )
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem coi1
StepHypRef Expression
1 relco 5169 . 2  |-  Rel  ( A  o.  _I  )
2 vex 2792 . . . . . 6  |-  x  e. 
_V
3 vex 2792 . . . . . 6  |-  y  e. 
_V
42, 3opelco 4852 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<->  E. z ( x  _I  z  /\  z A y ) )
5 vex 2792 . . . . . . . . . 10  |-  z  e. 
_V
65ideq 4835 . . . . . . . . 9  |-  ( x  _I  z  <->  x  =  z )
7 equcom 1648 . . . . . . . . 9  |-  ( x  =  z  <->  z  =  x )
86, 7bitri 242 . . . . . . . 8  |-  ( x  _I  z  <->  z  =  x )
98anbi1i 678 . . . . . . 7  |-  ( ( x  _I  z  /\  z A y )  <->  ( z  =  x  /\  z A y ) )
109exbii 1570 . . . . . 6  |-  ( E. z ( x  _I  z  /\  z A y )  <->  E. z
( z  =  x  /\  z A y ) )
11 breq1 4027 . . . . . . 7  |-  ( z  =  x  ->  (
z A y  <->  x A
y ) )
122, 11ceqsexv 2824 . . . . . 6  |-  ( E. z ( z  =  x  /\  z A y )  <->  x A
y )
1310, 12bitri 242 . . . . 5  |-  ( E. z ( x  _I  z  /\  z A y )  <->  x A
y )
144, 13bitri 242 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<->  x A y )
15 df-br 4025 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
1614, 15bitri 242 . . 3  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<-> 
<. x ,  y >.  e.  A )
1716eqrelriv 4779 . 2  |-  ( ( Rel  ( A  o.  _I  )  /\  Rel  A
)  ->  ( A  o.  _I  )  =  A )
181, 17mpan 653 1  |-  ( Rel 
A  ->  ( A  o.  _I  )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1529    = wceq 1624    e. wcel 1685   <.cop 3644   class class class wbr 4024    _I cid 4303    o. ccom 4692   Rel wrel 4693
This theorem is referenced by:  coi2  5187  coires1  5188  relcoi1  5199  fcoi1  5380  cocnv  25792  mvdco  26787
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-co 4697
  Copyright terms: Public domain W3C validator