Theorem coi2 5147

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
coi2	|- ( Rel A -> ( _I o. A ) = A )

Proof of Theorem coi2

Step	Hyp	Ref	Expression
1		cnvco 4814	. . 3 |- `' ( `' A o. _I ) = ( `' _I o. `' `' A )
2		relcnv 5008	. . . . 5 |- Rel `' A
3		coi1 5146	. . . . 5 |- ( Rel `' A -> ( `' A o. _I ) = `' A )
4	2, 3	ax-mp 5	. . . 4 |- ( `' A o. _I ) = `' A
5	4	cnveqi 4804	. . 3 |- `' ( `' A o. _I ) = `' `' A
6	1, 5	eqtr3i 2200	. 2 |- ( `' _I o. `' `' A ) = `' `' A
7		dfrel2 5081	. . 3 |- ( Rel A <-> `' `' A = A )
8		cnvi 5035	. . . 4 |- `' _I = _I
9		coeq2 4787	. . . . 5 |- ( `' `' A = A -> ( `' _I o. `' `' A ) = ( `' _I o. A ) )
10		coeq1 4786	. . . . 5 |- ( `' _I = _I -> ( `' _I o. A ) = ( _I o. A ) )
11	9, 10	sylan9eq 2230	. . . 4 |- ( ( `' `' A = A /\ `' _I = _I ) -> ( `' _I o. `' `' A ) = ( _I o. A ) )
12	8, 11	mpan2 425	. . 3 |- ( `' `' A = A -> ( `' _I o. `' `' A ) = ( _I o. A ) )
13	7, 12	sylbi 121	. 2 |- ( Rel A -> ( `' _I o. `' `' A ) = ( _I o. A ) )
14	7	biimpi 120	. 2 |- ( Rel A -> `' `' A = A )
15	6, 13, 14	3eqtr3a 2234	1 |- ( Rel A -> ( _I o. A ) = A )

Syntax hints:   -> wi 4   = wceq 1353   _I cid 4290   `'ccnv 4627   o. ccom 4632   Rel wrel 4633

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637

This theorem is referenced by:  relcoi2  5161  funi  5250  fcoi2  5399