Theorem coi1 5301

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 5284	. 2 |- Rel ( A o. _I )
2		vex 2824	. . . . . 6 |- x e. _V
3		vex 2824	. . . . . 6 |- y e. _V
4	2, 3	opelco 4950	. . . . 5 |- ( <. x , y >. e. ( A o. _I ) <-> E. z ( x _I z /\ z A y ) )
5		vex 2824	. . . . . . . . . 10 |- z e. _V
6	5	ideq 4930	. . . . . . . . 9 |- ( x _I z <-> x = z )
7		equcom 1758	. . . . . . . . 9 |- ( x = z <-> z = x )
8	6, 7	bitri 184	. . . . . . . 8 |- ( x _I z <-> z = x )
9	8	anbi1i 462	. . . . . . 7 |- ( ( x _I z /\ z A y ) <-> ( z = x /\ z A y ) )
10	9	exbii 1658	. . . . . 6 |- ( E. z ( x _I z /\ z A y ) <-> E. z ( z = x /\ z A y ) )
11		breq1 4131	. . . . . . 7 |- ( z = x -> ( z A y <-> x A y ) )
12	2, 11	ceqsexv 2861	. . . . . 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 4129	. . . 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 4866	. 2 |- ( ( Rel ( A o. _I ) /\ Rel A ) -> ( A o. _I ) = A )
18	1, 17	mpan 428	1 |- ( Rel A -> ( A o. _I ) = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   <.cop 3711   class class class wbr 4128   _I cid 4431   o. ccom 4776   Rel wrel 4777

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-co 4781

This theorem is referenced by:  coi2  5302  coires1  5303  relcoi1  5317  fcoi1  5570