Theorem coi2 5652

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		dfrel2 5583	. 2 |- ( Rel A <-> `' `' A = A )
2		cnvco 5308	. . . 4 |- `' ( `' A o. _I ) = ( `' _I o. `' `' A )
3		relcnv 5503	. . . . . 6 |- Rel `' A
4		coi1 5651	. . . . . 6 |- ( Rel `' A -> ( `' A o. _I ) = `' A )
5	3, 4	ax-mp 5	. . . . 5 |- ( `' A o. _I ) = `' A
6	5	cnveqi 5297	. . . 4 |- `' ( `' A o. _I ) = `' `' A
7	2, 6	eqtr3i 2646	. . 3 |- ( `' _I o. `' `' A ) = `' `' A
8		cnvi 5537	. . . 4 |- `' _I = _I
9		coeq2 5280	. . . . 5 |- ( `' `' A = A -> ( `' _I o. `' `' A ) = ( `' _I o. A ) )
10		coeq1 5279	. . . . 5 |- ( `' _I = _I -> ( `' _I o. A ) = ( _I o. A ) )
11	9, 10	sylan9eq 2676	. . . 4 |- ( ( `' `' A = A /\ `' _I = _I ) -> ( `' _I o. `' `' A ) = ( _I o. A ) )
12	8, 11	mpan2 707	. . 3 |- ( `' `' A = A -> ( `' _I o. `' `' A ) = ( _I o. A ) )
13		id 22	. . 3 |- ( `' `' A = A -> `' `' A = A )
14	7, 12, 13	3eqtr3a 2680	. 2 |- ( `' `' A = A -> ( _I o. A ) = A )
15	1, 14	sylbi 207	1 |- ( Rel A -> ( _I o. A ) = A )

Syntax hints:   -> wi 4   = wceq 1483   _I cid 5023   `'ccnv 5113   o. ccom 5118   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123

This theorem is referenced by:  relcoi2  5663  funi  5920  fcoi2  6079