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Theorem coi1 5094
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
StepHypRef Expression
1 relco 5077 . 2  |-  Rel  ( A  o.  _I  )
2 vex 2730 . . . . . 6  |-  x  e. 
_V
3 vex 2730 . . . . . 6  |-  y  e. 
_V
42, 3opelco 4760 . . . . 5  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<->  E. z ( x  _I  z  /\  z A y ) )
5 vex 2730 . . . . . . . . . 10  |-  z  e. 
_V
65ideq 4743 . . . . . . . . 9  |-  ( x  _I  z  <->  x  =  z )
7 equcom 1824 . . . . . . . . 9  |-  ( x  =  z  <->  z  =  x )
86, 7bitri 242 . . . . . . . 8  |-  ( x  _I  z  <->  z  =  x )
98anbi1i 679 . . . . . . 7  |-  ( ( x  _I  z  /\  z A y )  <->  ( z  =  x  /\  z A y ) )
109exbii 1580 . . . . . 6  |-  ( E. z ( x  _I  z  /\  z A y )  <->  E. z
( z  =  x  /\  z A y ) )
11 breq1 3923 . . . . . . 7  |-  ( z  =  x  ->  (
z A y  <->  x A
y ) )
122, 11ceqsexv 2761 . . . . . 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 3921 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
1614, 15bitri 242 . . 3  |-  ( <.
x ,  y >.  e.  ( A  o.  _I  ) 
<-> 
<. x ,  y >.  e.  A )
1716eqrelriv 4687 . 2  |-  ( ( Rel  ( A  o.  _I  )  /\  Rel  A
)  ->  ( A  o.  _I  )  =  A )
181, 17mpan 654 1  |-  ( Rel 
A  ->  ( A  o.  _I  )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   <.cop 3547   class class class wbr 3920    _I cid 4197    o. ccom 4584   Rel wrel 4585
This theorem is referenced by:  coi2  5095  coires1  5096  relcoi1  5107  fcoi1  5272  cocnv  25559  mvdco  26554
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-co 4597
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