Theorem coires1 4888

Description: Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)

Assertion

Ref	Expression
coires1	|- ( A o. ( _I |` B ) ) = ( A |` B )

Proof of Theorem coires1

Step	Hyp	Ref	Expression
1		cocnvcnv1 4881	. . . . 5 |- ( `' `' A o. _I ) = ( A o. _I )
2		relcnv 4753	. . . . . 6 |- Rel `' `' A
3		coi1 4886	. . . . . 6 |- ( Rel `' `' A -> ( `' `' A o. _I ) = `' `' A )
4	2, 3	ax-mp 7	. . . . 5 |- ( `' `' A o. _I ) = `' `' A
5	1, 4	eqtr3i 2105	. . . 4 |- ( A o. _I ) = `' `' A
6	5	reseq1i 4656	. . 3 |- ( ( A o. _I ) |` B ) = ( `' `' A |` B )
7		resco 4875	. . 3 |- ( ( A o. _I ) |` B ) = ( A o. ( _I |` B ) )
8	6, 7	eqtr3i 2105	. 2 |- ( `' `' A |` B ) = ( A o. ( _I |` B ) )
9		rescnvcnv 4833	. 2 |- ( `' `' A |` B ) = ( A |` B )
10	8, 9	eqtr3i 2105	1 |- ( A o. ( _I |` B ) ) = ( A |` B )

Syntax hints:   = wceq 1285   _I cid 4071   `'ccnv 4390   |` cres 4393   o. ccom 4395   Rel wrel 4396

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403

This theorem is referenced by:  funcoeqres  5208