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.

Step	Hyp	Ref	Expression
1		relco 5351	. 2 |- Rel ( A o. _I )
2		vex 3059	. . . . . 6 |- x e. _V
3		vex 3059	. . . . . 6 |- y e. _V
4	2, 3	opelco 5024	. . . . 5 |- ( <. x , y >. e. ( A o. _I ) <-> E. z ( x _I z /\ z A y ) )
5		vex 3059	. . . . . . . . . 10 |- z e. _V
6	5	ideq 5005	. . . . . . . . 9 |- ( x _I z <-> x = z )
7		equcom 1872	. . . . . . . . 9 |- ( x = z <-> z = x )
8	6, 7	bitri 257	. . . . . . . 8 |- ( x _I z <-> z = x )
9	8	anbi1i 706	. . . . . . 7 |- ( ( x _I z /\ z A y ) <-> ( z = x /\ z A y ) )
10	9	exbii 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 ) )
12	2, 11	ceqsexv 3095	. . . . . 6 |- ( E. z ( z = x /\ z A y ) <-> x A y )
13	10, 12	bitri 257	. . . . 5 |- ( E. z ( x _I z /\ z A y ) <-> x A y )
14	4, 13	bitri 257	. . . 4 |- ( <. x , y >. e. ( A o. _I ) <-> x A y )
15		df-br 4416	. . . 4 |- ( x A y <-> <. x , y >. e. A )
16	14, 15	bitri 257	. . 3 |- ( <. x , y >. e. ( A o. _I ) <-> <. x , y >. e. A )
17	16	eqrelriv 4946	. 2 |- ( ( Rel ( A o. _I ) /\ Rel A ) -> ( A o. _I ) = A )
18	1, 17	mpan 681	1 |- ( Rel A -> ( A o. _I ) = A )

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