Theorem coires1 5148

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 5141	. . . . 5 |- ( `' `' A o. _I ) = ( A o. _I )
2		relcnv 5008	. . . . . 6 |- Rel `' `' A
3		coi1 5146	. . . . . 6 |- ( Rel `' `' A -> ( `' `' A o. _I ) = `' `' A )
4	2, 3	ax-mp 5	. . . . 5 |- ( `' `' A o. _I ) = `' `' A
5	1, 4	eqtr3i 2200	. . . 4 |- ( A o. _I ) = `' `' A
6	5	reseq1i 4905	. . 3 |- ( ( A o. _I ) |` B ) = ( `' `' A |` B )
7		resco 5135	. . 3 |- ( ( A o. _I ) |` B ) = ( A o. ( _I |` B ) )
8	6, 7	eqtr3i 2200	. 2 |- ( `' `' A |` B ) = ( A o. ( _I |` B ) )
9		rescnvcnv 5093	. 2 |- ( `' `' A |` B ) = ( A |` B )
10	8, 9	eqtr3i 2200	1 |- ( A o. ( _I |` B ) ) = ( A |` B )

Syntax hints:   = wceq 1353   _I cid 4290   `'ccnv 4627   |` cres 4630   o. ccom 4632   Rel wrel 4633

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640

This theorem is referenced by:  funcoeqres  5494