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

Theorem coi1 5376
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 5359 . 2  |-  Rel  ( A  o.  _I  )
2 vex 2951 . . . . . 6  |-  x  e. 
_V
3 vex 2951 . . . . . 6  |-  y  e. 
_V
42, 3opelco 5035 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<->  E. z ( x  _I  z  /\  z A y ) )
5 vex 2951 . . . . . . . . . 10  |-  z  e. 
_V
65ideq 5016 . . . . . . . . 9  |-  ( x  _I  z  <->  x  =  z )
7 equcom 1692 . . . . . . . . 9  |-  ( x  =  z  <->  z  =  x )
86, 7bitri 241 . . . . . . . 8  |-  ( x  _I  z  <->  z  =  x )
98anbi1i 677 . . . . . . 7  |-  ( ( x  _I  z  /\  z A y )  <->  ( z  =  x  /\  z A y ) )
109exbii 1592 . . . . . 6  |-  ( E. z ( x  _I  z  /\  z A y )  <->  E. z
( z  =  x  /\  z A y ) )
11 breq1 4207 . . . . . . 7  |-  ( z  =  x  ->  (
z A y  <->  x A
y ) )
122, 11ceqsexv 2983 . . . . . 6  |-  ( E. z ( z  =  x  /\  z A y )  <->  x A
y )
1310, 12bitri 241 . . . . 5  |-  ( E. z ( x  _I  z  /\  z A y )  <->  x A
y )
144, 13bitri 241 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<->  x A y )
15 df-br 4205 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
1614, 15bitri 241 . . 3  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<-> 
<. x ,  y >.  e.  A )
1716eqrelriv 4960 . 2  |-  ( ( Rel  ( A  o.  _I  )  /\  Rel  A
)  ->  ( A  o.  _I  )  =  A )
181, 17mpan 652 1  |-  ( Rel 
A  ->  ( A  o.  _I  )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204    _I cid 4485    o. ccom 4873   Rel wrel 4874
This theorem is referenced by:  coi2  5377  coires1  5378  relcoi1  5389  fcoi1  5608  cocnv  26364  mvdco  27303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-co 4878
  Copyright terms: Public domain W3C validator