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Theorem fcoi1 5415
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 5389 . 2  |-  ( F : A --> B  ->  F  Fn  A )
2 df-fn 5258 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
3 eqimss 3230 . . . . 5  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
4 cnvi 5085 . . . . . . . . . 10  |-  `'  _I  =  _I
54reseq1i 4951 . . . . . . . . 9  |-  ( `'  _I  |`  A )  =  (  _I  |`  A )
65cnveqi 4856 . . . . . . . 8  |-  `' ( `'  _I  |`  A )  =  `' (  _I  |`  A )
7 cnvresid 5322 . . . . . . . 8  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
86, 7eqtr2i 2304 . . . . . . 7  |-  (  _I  |`  A )  =  `' ( `'  _I  |`  A )
98coeq2i 4844 . . . . . 6  |-  ( F  o.  (  _I  |`  A ) )  =  ( F  o.  `' ( `'  _I  |`  A )
)
10 cores2 5185 . . . . . 6  |-  ( dom 
F  C_  A  ->  ( F  o.  `' ( `'  _I  |`  A )
)  =  ( F  o.  _I  ) )
119, 10syl5eq 2327 . . . . 5  |-  ( dom 
F  C_  A  ->  ( F  o.  (  _I  |`  A ) )  =  ( F  o.  _I  ) )
123, 11syl 15 . . . 4  |-  ( dom 
F  =  A  -> 
( F  o.  (  _I  |`  A ) )  =  ( F  o.  _I  ) )
13 funrel 5272 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
14 coi1 5188 . . . . 5  |-  ( Rel 
F  ->  ( F  o.  _I  )  =  F )
1513, 14syl 15 . . . 4  |-  ( Fun 
F  ->  ( F  o.  _I  )  =  F )
1612, 15sylan9eqr 2337 . . 3  |-  ( ( Fun  F  /\  dom  F  =  A )  -> 
( F  o.  (  _I  |`  A ) )  =  F )
172, 16sylbi 187 . 2  |-  ( F  Fn  A  ->  ( F  o.  (  _I  |`  A ) )  =  F )
181, 17syl 15 1  |-  ( F : A --> B  -> 
( F  o.  (  _I  |`  A ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    C_ wss 3152    _I cid 4304   `'ccnv 4688   dom cdm 4689    |` cres 4691    o. ccom 4693   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  fcof1o  5803  mapen  7025  mapfien  7399  hashfacen  11392  cofurid  13765  setccatid  13916  curf2ndf  14021  symgid  14781  pf1mpf  19435  pf1ind  19438  wilthlem3  20308  hoico1  22336  crimmt1  25146  dispos  25287  cmpidmor3  25970  diophrw  26838  f1omvdco2  27391  psgnunilem1  27416  mendrng  27500  ltrncoidN  30317  trlcoabs2N  30911  trlcoat  30912  cdlemg47a  30923  cdlemg46  30924  trljco  30929  tendo1mulr  30960  tendo0co2  30977  cdlemi2  31008  cdlemk2  31021  cdlemk4  31023  cdlemk8  31027  cdlemk53  31146  cdlemk55a  31148  dvhopN  31306  dihopelvalcpre  31438  dihmeetlem1N  31480  dihglblem5apreN  31481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259
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