Theorem fcoi2 5312

Description: Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fcoi2	|- ( F : A --> B -> ( ( _I |` B ) o. F ) = F )

Proof of Theorem fcoi2

Step	Hyp	Ref	Expression
1		df-f 5135	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		cores 5050	. . 3 |- ( ran F C_ B -> ( ( _I |` B ) o. F ) = ( _I o. F ) )
3		fnrel 5229	. . . 4 |- ( F Fn A -> Rel F )
4		coi2 5063	. . . 4 |- ( Rel F -> ( _I o. F ) = F )
5	3, 4	syl 14	. . 3 |- ( F Fn A -> ( _I o. F ) = F )
6	2, 5	sylan9eqr 2195	. 2 |- ( ( F Fn A /\ ran F C_ B ) -> ( ( _I |` B ) o. F ) = F )
7	1, 6	sylbi 120	1 |- ( F : A --> B -> ( ( _I |` B ) o. F ) = F )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   C_ wss 3076   _I cid 4218   ran crn 4548   |` cres 4549   o. ccom 4551   Rel wrel 4552   Fn wfn 5126   -->wf 5127

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-fun 5133  df-fn 5134  df-f 5135

This theorem is referenced by:  fcof1o  5698  mapen  6748  hashfacen  10611