Theorem fcoi2 5515

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 5328	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		cores 5238	. . 3 |- ( ran F C_ B -> ( ( _I |` B ) o. F ) = ( _I o. F ) )
3		fnrel 5425	. . . 4 |- ( F Fn A -> Rel F )
4		coi2 5251	. . . 4 |- ( Rel F -> ( _I o. F ) = F )
5	3, 4	syl 14	. . 3 |- ( F Fn A -> ( _I o. F ) = F )
6	2, 5	sylan9eqr 2284	. 2 |- ( ( F Fn A /\ ran F C_ B ) -> ( ( _I |` B ) o. F ) = F )
7	1, 6	sylbi 121	1 |- ( F : A --> B -> ( ( _I |` B ) o. F ) = F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   C_ wss 3198   _I cid 4383   ran crn 4724   |` cres 4725   o. ccom 4727   Rel wrel 4728   Fn wfn 5319   -->wf 5320

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-fun 5326  df-fn 5327  df-f 5328

This theorem is referenced by:  fcof1o  5925  mapen  7027  hashfacen  11090