Theorem fcoi2 4586

Description: Composition of restricted identity and a mapping. (The proof was 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 4010	. 2 |- (F:A-->B <-> (F Fn A /\ ran F C_ B))
2		cores 4400	. . 3 |- (ran F C_ B -> (( _I |` B) o. F) = ( _I o. F))
3		fnrel 4511	. . . 4 |- (F Fn A -> Rel F)
4		coi2 4414	. . . 4 |- (Rel F -> ( _I o. F) = F)
5	3, 4	syl 12	. . 3 |- (F Fn A -> ( _I o. F) = F)
6	2, 5	sylan9eqr 1951	. 2 |- ((F Fn A /\ ran F C_ B) -> (( _I |` B) o. F) = F)
7	1, 6	sylbi 216	1 |- (F:A-->B -> (( _I |` B) o. F) = F)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   C_ wss 2593   _I cid 3582  ran crn 3987   |` cres 3988   o. ccom 3990  Rel wrel 3991   Fn wfn 3993  -->wf 3994

This theorem is referenced by:  mapenlem2 5584  symggrpi 10205  symgidi 10206  hoico2 11320  symgfo 14730  hmeogrp 14892

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-fun 4008  df-fn 4009  df-f 4010

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