Theorem coi1 5054

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.

Step	Hyp	Ref	Expression
1		relco 5037	. 2 |- Rel ( A o. _I )
2		vex 2689	. . . . . 6 |- x e. _V
3		vex 2689	. . . . . 6 |- y e. _V
4	2, 3	opelco 4711	. . . . 5 |- ( <. x , y >. e. ( A o. _I ) <-> E. z ( x _I z /\ z A y ) )
5		vex 2689	. . . . . . . . . 10 |- z e. _V
6	5	ideq 4691	. . . . . . . . 9 |- ( x _I z <-> x = z )
7		equcom 1682	. . . . . . . . 9 |- ( x = z <-> z = x )
8	6, 7	bitri 183	. . . . . . . 8 |- ( x _I z <-> z = x )
9	8	anbi1i 453	. . . . . . 7 |- ( ( x _I z /\ z A y ) <-> ( z = x /\ z A y ) )
10	9	exbii 1584	. . . . . 6 |- ( E. z ( x _I z /\ z A y ) <-> E. z ( z = x /\ z A y ) )
11		breq1 3932	. . . . . . 7 |- ( z = x -> ( z A y <-> x A y ) )
12	2, 11	ceqsexv 2725	. . . . . 6 |- ( E. z ( z = x /\ z A y ) <-> x A y )
13	10, 12	bitri 183	. . . . 5 |- ( E. z ( x _I z /\ z A y ) <-> x A y )
14	4, 13	bitri 183	. . . 4 |- ( <. x , y >. e. ( A o. _I ) <-> x A y )
15		df-br 3930	. . . 4 |- ( x A y <-> <. x , y >. e. A )
16	14, 15	bitri 183	. . 3 |- ( <. x , y >. e. ( A o. _I ) <-> <. x , y >. e. A )
17	16	eqrelriv 4632	. 2 |- ( ( Rel ( A o. _I ) /\ Rel A ) -> ( A o. _I ) = A )
18	1, 17	mpan 420	1 |- ( Rel A -> ( A o. _I ) = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   <.cop 3530   class class class wbr 3929   _I cid 4210   o. ccom 4543   Rel wrel 4544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-co 4548

This theorem is referenced by:  coi2  5055  coires1  5056  relcoi1  5070  fcoi1  5303