Theorem coi2 5186

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 4851	. . 3 |- `' ( `' A o. _I ) = ( `' _I o. `' `' A )
2		relcnv 5047	. . . . 5 |- Rel `' A
3		coi1 5185	. . . . 5 |- ( Rel `' A -> ( `' A o. _I ) = `' A )
4	2, 3	ax-mp 5	. . . 4 |- ( `' A o. _I ) = `' A
5	4	cnveqi 4841	. . 3 |- `' ( `' A o. _I ) = `' `' A
6	1, 5	eqtr3i 2219	. 2 |- ( `' _I o. `' `' A ) = `' `' A
7		dfrel2 5120	. . 3 |- ( Rel A <-> `' `' A = A )
8		cnvi 5074	. . . 4 |- `' _I = _I
9		coeq2 4824	. . . . 5 |- ( `' `' A = A -> ( `' _I o. `' `' A ) = ( `' _I o. A ) )
10		coeq1 4823	. . . . 5 |- ( `' _I = _I -> ( `' _I o. A ) = ( _I o. A ) )
11	9, 10	sylan9eq 2249	. . . 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 2253	1 |- ( Rel A -> ( _I o. A ) = A )

Syntax hints:   -> wi 4   = wceq 1364   _I cid 4323   `'ccnv 4662   o. ccom 4667   Rel wrel 4668

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672

This theorem is referenced by:  relcoi2  5200  funi  5290  fcoi2  5439