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Theorem fcoi2 5416
Description: Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fcoi2  |-  ( F : A --> B  -> 
( (  _I  |`  B )  o.  F )  =  F )

Proof of Theorem fcoi2
StepHypRef Expression
1 df-f 5259 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 cores 5176 . . 3  |-  ( ran 
F  C_  B  ->  ( (  _I  |`  B )  o.  F )  =  (  _I  o.  F
) )
3 fnrel 5342 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
4 coi2 5189 . . . 4  |-  ( Rel 
F  ->  (  _I  o.  F )  =  F )
53, 4syl 15 . . 3  |-  ( F  Fn  A  ->  (  _I  o.  F )  =  F )
62, 5sylan9eqr 2337 . 2  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  ->  ( (  _I  |`  B )  o.  F )  =  F )
71, 6sylbi 187 1  |-  ( F : A --> B  -> 
( (  _I  |`  B )  o.  F )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    C_ wss 3152    _I cid 4304   ran crn 4690    |` cres 4691    o. ccom 4693   Rel wrel 4694    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  fcof1o  5803  mapen  7025  mapfien  7399  hashfacen  11392  cofulid  13764  setccatid  13916  symggrp  14780  gsumval3  15191  gsumzf1o  15196  frgpcyg  16527  qtophmeo  17508  hoico2  22337  subfacp1lem5  23715  crimmt2  25147  hmeogrpi  25536  cmpidmor2  25969  f1linds  27295  f1omvdco2  27391  symggen  27411  psgnunilem1  27416  mendrng  27500  ltrncoidN  30317  trlcoat  30912  trlcone  30917  cdlemg47a  30923  cdlemg47  30925  trljco  30929  tgrpgrplem  30938  tendo1mul  30959  tendo0pl  30980  cdlemkid2  31113  cdlemk45  31136  cdlemk53b  31145  erng1r  31184  tendocnv  31211  dvalveclem  31215  dva0g  31217  dvhgrp  31297  dvhlveclem  31298  dvh0g  31301  cdlemn8  31394  dihordlem7b  31405  dihopelvalcpre  31438
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-fun 5257  df-fn 5258  df-f 5259
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