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Theorem coi1 5369
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 5351 . 2  |-  Rel  ( A  o.  _I  )
2 vex 3059 . . . . . 6  |-  x  e. 
_V
3 vex 3059 . . . . . 6  |-  y  e. 
_V
42, 3opelco 5024 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<->  E. z ( x  _I  z  /\  z A y ) )
5 vex 3059 . . . . . . . . . 10  |-  z  e. 
_V
65ideq 5005 . . . . . . . . 9  |-  ( x  _I  z  <->  x  =  z )
7 equcom 1872 . . . . . . . . 9  |-  ( x  =  z  <->  z  =  x )
86, 7bitri 257 . . . . . . . 8  |-  ( x  _I  z  <->  z  =  x )
98anbi1i 706 . . . . . . 7  |-  ( ( x  _I  z  /\  z A y )  <->  ( z  =  x  /\  z A y ) )
109exbii 1728 . . . . . 6  |-  ( E. z ( x  _I  z  /\  z A y )  <->  E. z
( z  =  x  /\  z A y ) )
11 breq1 4418 . . . . . . 7  |-  ( z  =  x  ->  (
z A y  <->  x A
y ) )
122, 11ceqsexv 3095 . . . . . 6  |-  ( E. z ( z  =  x  /\  z A y )  <->  x A
y )
1310, 12bitri 257 . . . . 5  |-  ( E. z ( x  _I  z  /\  z A y )  <->  x A
y )
144, 13bitri 257 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<->  x A y )
15 df-br 4416 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
1614, 15bitri 257 . . 3  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<-> 
<. x ,  y >.  e.  A )
1716eqrelriv 4946 . 2  |-  ( ( Rel  ( A  o.  _I  )  /\  Rel  A
)  ->  ( A  o.  _I  )  =  A )
181, 17mpan 681 1  |-  ( Rel 
A  ->  ( A  o.  _I  )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454   E.wex 1673    e. wcel 1897   <.cop 3985   class class class wbr 4415    _I cid 4762    o. ccom 4856   Rel wrel 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-co 4861
This theorem is referenced by:  coi2  5370  coires1  5371  relcoi1OLD  5383  fcoi1  5779  mvdco  17134  cocnv  32096
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