Theorem coi2 5244

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 4906	. . 3 |- `' ( `' A o. _I ) = ( `' _I o. `' `' A )
2		relcnv 5105	. . . . 5 |- Rel `' A
3		coi1 5243	. . . . 5 |- ( Rel `' A -> ( `' A o. _I ) = `' A )
4	2, 3	ax-mp 5	. . . 4 |- ( `' A o. _I ) = `' A
5	4	cnveqi 4896	. . 3 |- `' ( `' A o. _I ) = `' `' A
6	1, 5	eqtr3i 2252	. 2 |- ( `' _I o. `' `' A ) = `' `' A
7		dfrel2 5178	. . 3 |- ( Rel A <-> `' `' A = A )
8		cnvi 5132	. . . 4 |- `' _I = _I
9		coeq2 4879	. . . . 5 |- ( `' `' A = A -> ( `' _I o. `' `' A ) = ( `' _I o. A ) )
10		coeq1 4878	. . . . 5 |- ( `' _I = _I -> ( `' _I o. A ) = ( _I o. A ) )
11	9, 10	sylan9eq 2282	. . . 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 2286	1 |- ( Rel A -> ( _I o. A ) = A )

Syntax hints:   -> wi 4   = wceq 1395   _I cid 4378   `'ccnv 4717   o. ccom 4722   Rel wrel 4723

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727

This theorem is referenced by:  relcoi2  5258  funi  5349  fcoi2  5506