Theorem coi2 5102

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 4771	. . 3 |- `' ( `' A o. _I ) = ( `' _I o. `' `' A )
2		relcnv 4964	. . . . 5 |- Rel `' A
3		coi1 5101	. . . . 5 |- ( Rel `' A -> ( `' A o. _I ) = `' A )
4	2, 3	ax-mp 5	. . . 4 |- ( `' A o. _I ) = `' A
5	4	cnveqi 4761	. . 3 |- `' ( `' A o. _I ) = `' `' A
6	1, 5	eqtr3i 2180	. 2 |- ( `' _I o. `' `' A ) = `' `' A
7		dfrel2 5036	. . 3 |- ( Rel A <-> `' `' A = A )
8		cnvi 4990	. . . 4 |- `' _I = _I
9		coeq2 4744	. . . . 5 |- ( `' `' A = A -> ( `' _I o. `' `' A ) = ( `' _I o. A ) )
10		coeq1 4743	. . . . 5 |- ( `' _I = _I -> ( `' _I o. A ) = ( _I o. A ) )
11	9, 10	sylan9eq 2210	. . . 4 |- ( ( `' `' A = A /\ `' _I = _I ) -> ( `' _I o. `' `' A ) = ( _I o. A ) )
12	8, 11	mpan2 422	. . 3 |- ( `' `' A = A -> ( `' _I o. `' `' A ) = ( _I o. A ) )
13	7, 12	sylbi 120	. 2 |- ( Rel A -> ( `' _I o. `' `' A ) = ( _I o. A ) )
14	7	biimpi 119	. 2 |- ( Rel A -> `' `' A = A )
15	6, 13, 14	3eqtr3a 2214	1 |- ( Rel A -> ( _I o. A ) = A )

Syntax hints:   -> wi 4   = wceq 1335   _I cid 4248   `'ccnv 4585   o. ccom 4590   Rel wrel 4591

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595

This theorem is referenced by:  relcoi2  5116  funi  5202  fcoi2  5351