Theorem fcoi2 5369

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 5192	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		cores 5107	. . 3 |- ( ran F C_ B -> ( ( _I |` B ) o. F ) = ( _I o. F ) )
3		fnrel 5286	. . . 4 |- ( F Fn A -> Rel F )
4		coi2 5120	. . . 4 |- ( Rel F -> ( _I o. F ) = F )
5	3, 4	syl 14	. . 3 |- ( F Fn A -> ( _I o. F ) = F )
6	2, 5	sylan9eqr 2221	. 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 1343   C_ wss 3116   _I cid 4266   ran crn 4605   |` cres 4606   o. ccom 4608   Rel wrel 4609   Fn wfn 5183   -->wf 5184

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  fcof1o  5757  mapen  6812  hashfacen  10749