Theorem coires1 5056

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 5049	. . . . 5 |- ( `' `' A o. _I ) = ( A o. _I )
2		relcnv 4917	. . . . . 6 |- Rel `' `' A
3		coi1 5054	. . . . . 6 |- ( Rel `' `' A -> ( `' `' A o. _I ) = `' `' A )
4	2, 3	ax-mp 5	. . . . 5 |- ( `' `' A o. _I ) = `' `' A
5	1, 4	eqtr3i 2162	. . . 4 |- ( A o. _I ) = `' `' A
6	5	reseq1i 4815	. . 3 |- ( ( A o. _I ) |` B ) = ( `' `' A |` B )
7		resco 5043	. . 3 |- ( ( A o. _I ) |` B ) = ( A o. ( _I |` B ) )
8	6, 7	eqtr3i 2162	. 2 |- ( `' `' A |` B ) = ( A o. ( _I |` B ) )
9		rescnvcnv 5001	. 2 |- ( `' `' A |` B ) = ( A |` B )
10	8, 9	eqtr3i 2162	1 |- ( A o. ( _I |` B ) ) = ( A |` B )

Syntax hints:   = wceq 1331   _I cid 4210   `'ccnv 4538   |` cres 4541   o. ccom 4543   Rel wrel 4544

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551

This theorem is referenced by:  funcoeqres  5398