Theorem fcoi1 5311

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 5280	. 2 |- ( F : A --> B -> F Fn A )
2		df-fn 5134	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
3		eqimss 3156	. . . . 5 |- ( dom F = A -> dom F C_ A )
4		cnvi 4951	. . . . . . . . . 10 |- `' _I = _I
5	4	reseq1i 4823	. . . . . . . . 9 |- ( `' _I |` A ) = ( _I |` A )
6	5	cnveqi 4722	. . . . . . . 8 |- `' ( `' _I |` A ) = `' ( _I |` A )
7		cnvresid 5205	. . . . . . . 8 |- `' ( _I |` A ) = ( _I |` A )
8	6, 7	eqtr2i 2162	. . . . . . 7 |- ( _I |` A ) = `' ( `' _I |` A )
9	8	coeq2i 4707	. . . . . 6 |- ( F o. ( _I |` A ) ) = ( F o. `' ( `' _I |` A ) )
10		cores2 5059	. . . . . 6 |- ( dom F C_ A -> ( F o. `' ( `' _I |` A ) ) = ( F o. _I ) )
11	9, 10	syl5eq 2185	. . . . 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 5148	. . . . 5 |- ( Fun F -> Rel F )
14		coi1 5062	. . . . 5 |- ( Rel F -> ( F o. _I ) = F )
15	13, 14	syl 14	. . . 4 |- ( Fun F -> ( F o. _I ) = F )
16	12, 15	sylan9eqr 2195	. . 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 1332   C_ wss 3076   _I cid 4218   `'ccnv 4546   dom cdm 4547   |` cres 4549   o. ccom 4551   Rel wrel 4552   Fun wfun 5125   Fn wfn 5126   -->wf 5127

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-fun 5133  df-fn 5134  df-f 5135

This theorem is referenced by:  fcof1o  5698  mapen  6748  hashfacen  10611