Theorem fcoi1 5547

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 5508	. 2 |- ( F : A --> B -> F Fn A )
2		df-fn 5355	. . 3 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
3		eqimss 3292	. . . . 5 |- ( dom F = A -> dom F C_ A )
4		cnvi 5167	. . . . . . . . . 10 |- `' _I = _I
5	4	reseq1i 5034	. . . . . . . . 9 |- ( `' _I |` A ) = ( _I |` A )
6	5	cnveqi 4930	. . . . . . . 8 |- `' ( `' _I |` A ) = `' ( _I |` A )
7		cnvresid 5430	. . . . . . . 8 |- `' ( _I |` A ) = ( _I |` A )
8	6, 7	eqtr2i 2254	. . . . . . 7 |- ( _I |` A ) = `' ( `' _I |` A )
9	8	coeq2i 4915	. . . . . 6 |- ( F o. ( _I |` A ) ) = ( F o. `' ( `' _I |` A ) )
10		cores2 5275	. . . . . 6 |- ( dom F C_ A -> ( F o. `' ( `' _I |` A ) ) = ( F o. _I ) )
11	9, 10	eqtrid 2277	. . . . 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 5369	. . . . 5 |- ( Fun F -> Rel F )
14		coi1 5278	. . . . 5 |- ( Rel F -> ( F o. _I ) = F )
15	13, 14	syl 14	. . . 4 |- ( Fun F -> ( F o. _I ) = F )
16	12, 15	sylan9eqr 2287	. . 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 3211   _I cid 4409   `'ccnv 4748   dom cdm 4749   |` cres 4751   o. ccom 4753   Rel wrel 4754   Fun wfun 5346   Fn wfn 5347   -->wf 5348

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-fun 5354  df-fn 5355  df-f 5356

This theorem is referenced by:  fcof1o  5962  mapen  7099  hashfacen  11208  gsumgfsum1  16863