Theorem coi1 5181

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 5164	. 2 |- Rel ( A o. _I )
2		vex 2763	. . . . . 6 |- x e. _V
3		vex 2763	. . . . . 6 |- y e. _V
4	2, 3	opelco 4834	. . . . 5 |- ( <. x , y >. e. ( A o. _I ) <-> E. z ( x _I z /\ z A y ) )
5		vex 2763	. . . . . . . . . 10 |- z e. _V
6	5	ideq 4814	. . . . . . . . 9 |- ( x _I z <-> x = z )
7		equcom 1717	. . . . . . . . 9 |- ( x = z <-> z = x )
8	6, 7	bitri 184	. . . . . . . 8 |- ( x _I z <-> z = x )
9	8	anbi1i 458	. . . . . . 7 |- ( ( x _I z /\ z A y ) <-> ( z = x /\ z A y ) )
10	9	exbii 1616	. . . . . 6 |- ( E. z ( x _I z /\ z A y ) <-> E. z ( z = x /\ z A y ) )
11		breq1 4032	. . . . . . 7 |- ( z = x -> ( z A y <-> x A y ) )
12	2, 11	ceqsexv 2799	. . . . . 6 |- ( E. z ( z = x /\ z A y ) <-> x A y )
13	10, 12	bitri 184	. . . . 5 |- ( E. z ( x _I z /\ z A y ) <-> x A y )
14	4, 13	bitri 184	. . . 4 |- ( <. x , y >. e. ( A o. _I ) <-> x A y )
15		df-br 4030	. . . 4 |- ( x A y <-> <. x , y >. e. A )
16	14, 15	bitri 184	. . 3 |- ( <. x , y >. e. ( A o. _I ) <-> <. x , y >. e. A )
17	16	eqrelriv 4752	. 2 |- ( ( Rel ( A o. _I ) /\ Rel A ) -> ( A o. _I ) = A )
18	1, 17	mpan 424	1 |- ( Rel A -> ( A o. _I ) = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1503   e. wcel 2164   <.cop 3621   class class class wbr 4029   _I cid 4319   o. ccom 4663   Rel wrel 4664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-co 4668

This theorem is referenced by:  coi2  5182  coires1  5183  relcoi1  5197  fcoi1  5434