Theorem fcoi1 5368

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 5337	. 2 |- ( F : A --> B -> F Fn A )
2		df-fn 5191	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
3		eqimss 3196	. . . . 5 |- ( dom F = A -> dom F C_ A )
4		cnvi 5008	. . . . . . . . . 10 |- `' _I = _I
5	4	reseq1i 4880	. . . . . . . . 9 |- ( `' _I |` A ) = ( _I |` A )
6	5	cnveqi 4779	. . . . . . . 8 |- `' ( `' _I |` A ) = `' ( _I |` A )
7		cnvresid 5262	. . . . . . . 8 |- `' ( _I |` A ) = ( _I |` A )
8	6, 7	eqtr2i 2187	. . . . . . 7 |- ( _I |` A ) = `' ( `' _I |` A )
9	8	coeq2i 4764	. . . . . 6 |- ( F o. ( _I |` A ) ) = ( F o. `' ( `' _I |` A ) )
10		cores2 5116	. . . . . 6 |- ( dom F C_ A -> ( F o. `' ( `' _I |` A ) ) = ( F o. _I ) )
11	9, 10	syl5eq 2211	. . . . 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 5205	. . . . 5 |- ( Fun F -> Rel F )
14		coi1 5119	. . . . 5 |- ( Rel F -> ( F o. _I ) = F )
15	13, 14	syl 14	. . . 4 |- ( Fun F -> ( F o. _I ) = F )
16	12, 15	sylan9eqr 2221	. . 3 |- ( ( Fun F /\ dom F = A ) -> ( F o. ( _I |` A ) ) = F )
17	2, 16	sylbi 120	. 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 103   = wceq 1343   C_ wss 3116   _I cid 4266   `'ccnv 4603   dom cdm 4604   |` cres 4606   o. ccom 4608   Rel wrel 4609   Fun wfun 5182   Fn wfn 5183   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  fcof1o  5757  mapen  6812  hashfacen  10749