Theorem fcoi2 5436

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 5259	. 2 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		cores 5170	. . 3 |- ( ran F C_ B -> ( ( _I |` B ) o. F ) = ( _I o. F ) )
3		fnrel 5353	. . . 4 |- ( F Fn A -> Rel F )
4		coi2 5183	. . . 4 |- ( Rel F -> ( _I o. F ) = F )
5	3, 4	syl 14	. . 3 |- ( F Fn A -> ( _I o. F ) = F )
6	2, 5	sylan9eqr 2248	. 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 1364   C_ wss 3154   _I cid 4320   ran crn 4661   |` cres 4662   o. ccom 4664   Rel wrel 4665   Fn wfn 5250   -->wf 5251

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-fun 5257  df-fn 5258  df-f 5259

This theorem is referenced by:  fcof1o  5833  mapen  6904  hashfacen  10910