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Theorem fcoi1 5683
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
StepHypRef Expression
1 ffn 5657 . 2  |-  ( F : A --> B  ->  F  Fn  A )
2 df-fn 5519 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
3 eqimss 3506 . . . . 5  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
4 cnvi 5339 . . . . . . . . . 10  |-  `'  _I  =  _I
54reseq1i 5204 . . . . . . . . 9  |-  ( `'  _I  |`  A )  =  (  _I  |`  A )
65cnveqi 5112 . . . . . . . 8  |-  `' ( `'  _I  |`  A )  =  `' (  _I  |`  A )
7 cnvresid 5586 . . . . . . . 8  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
86, 7eqtr2i 2481 . . . . . . 7  |-  (  _I  |`  A )  =  `' ( `'  _I  |`  A )
98coeq2i 5098 . . . . . 6  |-  ( F  o.  (  _I  |`  A ) )  =  ( F  o.  `' ( `'  _I  |`  A )
)
10 cores2 5448 . . . . . 6  |-  ( dom 
F  C_  A  ->  ( F  o.  `' ( `'  _I  |`  A )
)  =  ( F  o.  _I  ) )
119, 10syl5eq 2504 . . . . 5  |-  ( dom 
F  C_  A  ->  ( F  o.  (  _I  |`  A ) )  =  ( F  o.  _I  ) )
123, 11syl 16 . . . 4  |-  ( dom 
F  =  A  -> 
( F  o.  (  _I  |`  A ) )  =  ( F  o.  _I  ) )
13 funrel 5533 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
14 coi1 5451 . . . . 5  |-  ( Rel 
F  ->  ( F  o.  _I  )  =  F )
1513, 14syl 16 . . . 4  |-  ( Fun 
F  ->  ( F  o.  _I  )  =  F )
1612, 15sylan9eqr 2514 . . 3  |-  ( ( Fun  F  /\  dom  F  =  A )  -> 
( F  o.  (  _I  |`  A ) )  =  F )
172, 16sylbi 195 . 2  |-  ( F  Fn  A  ->  ( F  o.  (  _I  |`  A ) )  =  F )
181, 17syl 16 1  |-  ( F : A --> B  -> 
( F  o.  (  _I  |`  A ) )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    C_ wss 3426    _I cid 4729   `'ccnv 4937   dom cdm 4938    |` cres 4940    o. ccom 4942   Rel wrel 4943   Fun wfun 5510    Fn wfn 5511   -->wf 5512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-fun 5518  df-fn 5519  df-f 5520
This theorem is referenced by:  fcof1o  6096  mapen  7575  mapfien  7758  mapfienOLD  8028  hashfacen  12309  cofurid  14903  setccatid  15054  curf2ndf  15159  symgid  16008  f1omvdco2  16056  psgnunilem1  16101  pf1mpf  17895  pf1ind  17898  wilthlem3  22524  hoico1  25295  fcobijfs  26160  diophrw  29235  mendrng  29687  ltrncoidN  34078  trlcoabs2N  34672  trlcoat  34673  cdlemg47a  34684  cdlemg46  34685  trljco  34690  tendo1mulr  34721  tendo0co2  34738  cdlemi2  34769  cdlemk2  34782  cdlemk4  34784  cdlemk8  34788  cdlemk53  34907  cdlemk55a  34909  dvhopN  35067  dihopelvalcpre  35199  dihmeetlem1N  35241  dihglblem5apreN  35242
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