Theorem fcoi1 5514

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 5479	. 2 |- ( F : A --> B -> F Fn A )
2		df-fn 5327	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
3		eqimss 3279	. . . . 5 |- ( dom F = A -> dom F C_ A )
4		cnvi 5139	. . . . . . . . . 10 |- `' _I = _I
5	4	reseq1i 5007	. . . . . . . . 9 |- ( `' _I |` A ) = ( _I |` A )
6	5	cnveqi 4903	. . . . . . . 8 |- `' ( `' _I |` A ) = `' ( _I |` A )
7		cnvresid 5401	. . . . . . . 8 |- `' ( _I |` A ) = ( _I |` A )
8	6, 7	eqtr2i 2251	. . . . . . 7 |- ( _I |` A ) = `' ( `' _I |` A )
9	8	coeq2i 4888	. . . . . 6 |- ( F o. ( _I |` A ) ) = ( F o. `' ( `' _I |` A ) )
10		cores2 5247	. . . . . 6 |- ( dom F C_ A -> ( F o. `' ( `' _I |` A ) ) = ( F o. _I ) )
11	9, 10	eqtrid 2274	. . . . 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 5341	. . . . 5 |- ( Fun F -> Rel F )
14		coi1 5250	. . . . 5 |- ( Rel F -> ( F o. _I ) = F )
15	13, 14	syl 14	. . . 4 |- ( Fun F -> ( F o. _I ) = F )
16	12, 15	sylan9eqr 2284	. . 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 1395   C_ wss 3198   _I cid 4383   `'ccnv 4722   dom cdm 4723   |` cres 4725   o. ccom 4727   Rel wrel 4728   Fun wfun 5318   Fn wfn 5319   -->wf 5320

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-fun 5326  df-fn 5327  df-f 5328

This theorem is referenced by:  fcof1o  5925  mapen  7027  hashfacen  11090