Theorem coi1 5651

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 5633	. 2 |- Rel ( A o. _I )
2		vex 3203	. . . . . 6 |- x e. _V
3		vex 3203	. . . . . 6 |- y e. _V
4	2, 3	opelco 5293	. . . . 5 |- ( <. x , y >. e. ( A o. _I ) <-> E. z ( x _I z /\ z A y ) )
5		vex 3203	. . . . . . . . . 10 |- z e. _V
6	5	ideq 5274	. . . . . . . . 9 |- ( x _I z <-> x = z )
7		equcom 1945	. . . . . . . . 9 |- ( x = z <-> z = x )
8	6, 7	bitri 264	. . . . . . . 8 |- ( x _I z <-> z = x )
9	8	anbi1i 731	. . . . . . 7 |- ( ( x _I z /\ z A y ) <-> ( z = x /\ z A y ) )
10	9	exbii 1774	. . . . . 6 |- ( E. z ( x _I z /\ z A y ) <-> E. z ( z = x /\ z A y ) )
11		breq1 4656	. . . . . . 7 |- ( z = x -> ( z A y <-> x A y ) )
12	11	equsexvw 1932	. . . . . 6 |- ( E. z ( z = x /\ z A y ) <-> x A y )
13	10, 12	bitri 264	. . . . 5 |- ( E. z ( x _I z /\ z A y ) <-> x A y )
14	4, 13	bitri 264	. . . 4 |- ( <. x , y >. e. ( A o. _I ) <-> x A y )
15		df-br 4654	. . . 4 |- ( x A y <-> <. x , y >. e. A )
16	14, 15	bitri 264	. . 3 |- ( <. x , y >. e. ( A o. _I ) <-> <. x , y >. e. A )
17	16	eqrelriv 5213	. 2 |- ( ( Rel ( A o. _I ) /\ Rel A ) -> ( A o. _I ) = A )
18	1, 17	mpan 706	1 |- ( Rel A -> ( A o. _I ) = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   <.cop 4183   class class class wbr 4653   _I cid 5023   o. ccom 5118   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-co 5123

This theorem is referenced by:  coi2  5652  coires1  5653  fcoi1  6078  mvdco  17865  cocnv  33520