Theorem fcoi1 5525

Description: Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof 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 5489	. 2 |- ( F : A --> B -> F Fn A )
2		df-fn 5336	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
3		eqimss 3282	. . . . 5 |- ( dom F = A -> dom F C_ A )
4		cnvi 5148	. . . . . . . . . 10 |- `' _I = _I
5	4	reseq1i 5015	. . . . . . . . 9 |- ( `' _I |` A ) = ( _I |` A )
6	5	cnveqi 4911	. . . . . . . 8 |- `' ( `' _I |` A ) = `' ( _I |` A )
7		cnvresid 5411	. . . . . . . 8 |- `' ( _I |` A ) = ( _I |` A )
8	6, 7	eqtr2i 2253	. . . . . . 7 |- ( _I |` A ) = `' ( `' _I |` A )
9	8	coeq2i 4896	. . . . . 6 |- ( F o. ( _I |` A ) ) = ( F o. `' ( `' _I |` A ) )
10		cores2 5256	. . . . . 6 |- ( dom F C_ A -> ( F o. `' ( `' _I |` A ) ) = ( F o. _I ) )
11	9, 10	eqtrid 2276	. . . . 5 |- ( dom F C_ A -> ( F o. ( _I |` A ) ) = ( F o. _I ) )
12	3, 11	syl 14	. . . 4 |- ( dom F = A -> ( F o. ( _I |` A ) ) = ( F o. _I ) )
13		funrel 5350	. . . . 5 |- ( Fun F -> Rel F )
14		coi1 5259	. . . . 5 |- ( Rel F -> ( F o. _I ) = F )
15	13, 14	syl 14	. . . 4 |- ( Fun F -> ( F o. _I ) = F )
16	12, 15	sylan9eqr 2286	. . 3 |- ( ( Fun F /\ dom F = A ) -> ( F o. ( _I |` A ) ) = F )
17	2, 16	sylbi 121	. 2 |- ( F Fn A -> ( F o. ( _I |` A ) ) = F )
18	1, 17	syl 14	1 |- ( F : A --> B -> ( F o. ( _I |` A ) ) = F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   C_ wss 3201   _I cid 4391   `'ccnv 4730   dom cdm 4731   |` cres 4733   o. ccom 4735   Rel wrel 4736   Fun wfun 5327   Fn wfn 5328   -->wf 5329

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-fun 5335  df-fn 5336  df-f 5337

This theorem is referenced by:  fcof1o  5940  mapen  7075  hashfacen  11163  gsumgfsum1  16810