Theorem coires1 5219

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 5212	. . . . 5 |- ( `' `' A o. _I ) = ( A o. _I )
2		relcnv 5079	. . . . . 6 |- Rel `' `' A
3		coi1 5217	. . . . . 6 |- ( Rel `' `' A -> ( `' `' A o. _I ) = `' `' A )
4	2, 3	ax-mp 5	. . . . 5 |- ( `' `' A o. _I ) = `' `' A
5	1, 4	eqtr3i 2230	. . . 4 |- ( A o. _I ) = `' `' A
6	5	reseq1i 4974	. . 3 |- ( ( A o. _I ) |` B ) = ( `' `' A |` B )
7		resco 5206	. . 3 |- ( ( A o. _I ) |` B ) = ( A o. ( _I |` B ) )
8	6, 7	eqtr3i 2230	. 2 |- ( `' `' A |` B ) = ( A o. ( _I |` B ) )
9		rescnvcnv 5164	. 2 |- ( `' `' A |` B ) = ( A |` B )
10	8, 9	eqtr3i 2230	1 |- ( A o. ( _I |` B ) ) = ( A |` B )

Syntax hints:   = wceq 1373   _I cid 4353   `'ccnv 4692   |` cres 4695   o. ccom 4697   Rel wrel 4698

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705

This theorem is referenced by:  funcoeqres  5575