Theorem coires1 4858

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 4851	. . . . 5 |- ( `' `' A o. _I ) = ( A o. _I )
2		relcnv 4723	. . . . . 6 |- Rel `' `' A
3		coi1 4856	. . . . . 6 |- ( Rel `' `' A -> ( `' `' A o. _I ) = `' `' A )
4	2, 3	ax-mp 7	. . . . 5 |- ( `' `' A o. _I ) = `' `' A
5	1, 4	eqtr3i 2103	. . . 4 |- ( A o. _I ) = `' `' A
6	5	reseq1i 4626	. . 3 |- ( ( A o. _I ) |` B ) = ( `' `' A |` B )
7		resco 4845	. . 3 |- ( ( A o. _I ) |` B ) = ( A o. ( _I |` B ) )
8	6, 7	eqtr3i 2103	. 2 |- ( `' `' A |` B ) = ( A o. ( _I |` B ) )
9		rescnvcnv 4803	. 2 |- ( `' `' A |` B ) = ( A |` B )
10	8, 9	eqtr3i 2103	1 |- ( A o. ( _I |` B ) ) = ( A |` B )

Syntax hints:   = wceq 1284   _I cid 4043   `'ccnv 4362   |` cres 4365   o. ccom 4367   Rel wrel 4368

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375

This theorem is referenced by:  funcoeqres  5177