Theorem fcoi1 4584

Description: Composition of a mapping and restricted identity. (The proof was shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fcoi1	|- (F:A-->B -> (F o. ( _I |` A)) = F)

Proof of Theorem fcoi1

Step	Hyp	Ref	Expression
1		ffn 4562	. 2 |- (F:A-->B -> F Fn A)
2		df-fn 4009	. . 3 |- (F Fn A <-> (Fun F /\ dom F = A))
3		eqimss 2665	. . . . 5 |- (dom F = A -> dom F C_ A)
4		cores2 4410	. . . . . 6 |- (dom F C_ A -> (F o. `'(`' _I |` A)) = (F o. _I ))
5		cnvi 4320	. . . . . . . . . 10 |- `' _I = _I
6		reseq1 4218	. . . . . . . . . 10 |- (`' _I = _I -> (`' _I |` A) = ( _I |` A))
7	5, 6	ax-mp 7	. . . . . . . . 9 |- (`' _I |` A) = ( _I |` A)
8	7	cnveqi 4136	. . . . . . . 8 |- `'(`' _I |` A) = `'( _I |` A)
9		cnvresid 4488	. . . . . . . 8 |- `'( _I |` A) = ( _I |` A)
10	8, 9	eqtr2i 1909	. . . . . . 7 |- ( _I |` A) = `'(`' _I |` A)
11	10	coeq2i 4126	. . . . . 6 |- (F o. ( _I |` A)) = (F o. `'(`' _I |` A))
12	4, 11	syl5eq 1940	. . . . 5 |- (dom F C_ A -> (F o. ( _I |` A)) = (F o. _I ))
13	3, 12	syl 12	. . . 4 |- (dom F = A -> (F o. ( _I |` A)) = (F o. _I ))
14		funrel 4438	. . . . 5 |- (Fun F -> Rel F)
15		coi1 4413	. . . . 5 |- (Rel F -> (F o. _I ) = F)
16	14, 15	syl 12	. . . 4 |- (Fun F -> (F o. _I ) = F)
17	13, 16	sylan9eqr 1951	. . 3 |- ((Fun F /\ dom F = A) -> (F o. ( _I |` A)) = F)
18	2, 17	sylbi 216	. 2 |- (F Fn A -> (F o. ( _I |` A)) = F)
19	1, 18	syl 12	1 |- (F:A-->B -> (F o. ( _I |` A)) = F)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   C_ wss 2593   _I cid 3582  `'ccnv 3985  dom cdm 3986   |` cres 3988   o. ccom 3990  Rel wrel 3991  Fun wfun 3992   Fn wfn 3993  -->wf 3994

This theorem is referenced by:  mapenlem1 5583  mapenlem2 5584  hoico1 11319  dispos 14632  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-rex 2110  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-ima 4007  df-fun 4008  df-fn 4009  df-f 4010

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