Theorem coi1 5126

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 5109	. 2 |- Rel ( A o. _I )
2		vex 2733	. . . . . 6 |- x e. _V
3		vex 2733	. . . . . 6 |- y e. _V
4	2, 3	opelco 4783	. . . . 5 |- ( <. x , y >. e. ( A o. _I ) <-> E. z ( x _I z /\ z A y ) )
5		vex 2733	. . . . . . . . . 10 |- z e. _V
6	5	ideq 4763	. . . . . . . . 9 |- ( x _I z <-> x = z )
7		equcom 1699	. . . . . . . . 9 |- ( x = z <-> z = x )
8	6, 7	bitri 183	. . . . . . . 8 |- ( x _I z <-> z = x )
9	8	anbi1i 455	. . . . . . 7 |- ( ( x _I z /\ z A y ) <-> ( z = x /\ z A y ) )
10	9	exbii 1598	. . . . . 6 |- ( E. z ( x _I z /\ z A y ) <-> E. z ( z = x /\ z A y ) )
11		breq1 3992	. . . . . . 7 |- ( z = x -> ( z A y <-> x A y ) )
12	2, 11	ceqsexv 2769	. . . . . 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 3990	. . . 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 4704	. 2 |- ( ( Rel ( A o. _I ) /\ Rel A ) -> ( A o. _I ) = A )
18	1, 17	mpan 422	1 |- ( Rel A -> ( A o. _I ) = A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   <.cop 3586   class class class wbr 3989   _I cid 4273   o. ccom 4615   Rel wrel 4616

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-co 4620

This theorem is referenced by:  coi2  5127  coires1  5128  relcoi1  5142  fcoi1  5378