Theorem coi2 5120

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 4789	. . 3 |- `' ( `' A o. _I ) = ( `' _I o. `' `' A )
2		relcnv 4982	. . . . 5 |- Rel `' A
3		coi1 5119	. . . . 5 |- ( Rel `' A -> ( `' A o. _I ) = `' A )
4	2, 3	ax-mp 5	. . . 4 |- ( `' A o. _I ) = `' A
5	4	cnveqi 4779	. . 3 |- `' ( `' A o. _I ) = `' `' A
6	1, 5	eqtr3i 2188	. 2 |- ( `' _I o. `' `' A ) = `' `' A
7		dfrel2 5054	. . 3 |- ( Rel A <-> `' `' A = A )
8		cnvi 5008	. . . 4 |- `' _I = _I
9		coeq2 4762	. . . . 5 |- ( `' `' A = A -> ( `' _I o. `' `' A ) = ( `' _I o. A ) )
10		coeq1 4761	. . . . 5 |- ( `' _I = _I -> ( `' _I o. A ) = ( _I o. A ) )
11	9, 10	sylan9eq 2219	. . . 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 2223	1 |- ( Rel A -> ( _I o. A ) = A )

Syntax hints:   -> wi 4   = wceq 1343   _I cid 4266   `'ccnv 4603   o. ccom 4608   Rel wrel 4609

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613

This theorem is referenced by:  relcoi2  5134  funi  5220  fcoi2  5369