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Theorem fcoi2 5647
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 5487 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 cores 5402 . . 3  |-  ( ran 
F  C_  B  ->  ( (  _I  |`  B )  o.  F )  =  (  _I  o.  F
) )
3 fnrel 5572 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
4 coi2 5415 . . . 4  |-  ( Rel 
F  ->  (  _I  o.  F )  =  F )
53, 4syl 16 . . 3  |-  ( F  Fn  A  ->  (  _I  o.  F )  =  F )
62, 5sylan9eqr 2496 . 2  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  ->  ( (  _I  |`  B )  o.  F )  =  F )
71, 6sylbi 189 1  |-  ( F : A --> B  -> 
( (  _I  |`  B )  o.  F )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    C_ wss 3306    _I cid 4522   ran crn 4908    |` cres 4909    o. ccom 4911   Rel wrel 4912    Fn wfn 5478   -->wf 5479
This theorem is referenced by:  fcof1o  6055  mapen  7300  mapfien  7682  hashfacen  11734  cofulid  14118  setccatid  14270  symggrp  15134  gsumval3  15545  gsumzf1o  15550  frgpcyg  16885  qtophmeo  17880  hoico2  23291  subfacp1lem5  24901  f1linds  27310  f1omvdco2  27406  symggen  27426  psgnunilem1  27431  mendrng  27515  ltrncoidN  31023  trlcoat  31618  trlcone  31623  cdlemg47a  31629  cdlemg47  31631  trljco  31635  tgrpgrplem  31644  tendo1mul  31665  tendo0pl  31686  cdlemkid2  31819  cdlemk45  31842  cdlemk53b  31851  erng1r  31890  tendocnv  31917  dvalveclem  31921  dva0g  31923  dvhgrp  32003  dvhlveclem  32004  dvh0g  32007  cdlemn8  32100  dihordlem7b  32111  dihopelvalcpre  32144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-fun 5485  df-fn 5486  df-f 5487
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