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Theorem fcoi2 5577
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 5417 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 cores 5332 . . 3  |-  ( ran 
F  C_  B  ->  ( (  _I  |`  B )  o.  F )  =  (  _I  o.  F
) )
3 fnrel 5502 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
4 coi2 5345 . . . 4  |-  ( Rel 
F  ->  (  _I  o.  F )  =  F )
53, 4syl 16 . . 3  |-  ( F  Fn  A  ->  (  _I  o.  F )  =  F )
62, 5sylan9eqr 2458 . 2  |-  ( ( F  Fn  A  /\  ran  F  C_  B )  ->  ( (  _I  |`  B )  o.  F )  =  F )
71, 6sylbi 188 1  |-  ( F : A --> B  -> 
( (  _I  |`  B )  o.  F )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    C_ wss 3280    _I cid 4453   ran crn 4838    |` cres 4839    o. ccom 4841   Rel wrel 4842    Fn wfn 5408   -->wf 5409
This theorem is referenced by:  fcof1o  5985  mapen  7230  mapfien  7609  hashfacen  11658  cofulid  14042  setccatid  14194  symggrp  15058  gsumval3  15469  gsumzf1o  15474  frgpcyg  16809  qtophmeo  17802  hoico2  23213  subfacp1lem5  24823  f1linds  27163  f1omvdco2  27259  symggen  27279  psgnunilem1  27284  mendrng  27368  ltrncoidN  30610  trlcoat  31205  trlcone  31210  cdlemg47a  31216  cdlemg47  31218  trljco  31222  tgrpgrplem  31231  tendo1mul  31252  tendo0pl  31273  cdlemkid2  31406  cdlemk45  31429  cdlemk53b  31438  erng1r  31477  tendocnv  31504  dvalveclem  31508  dva0g  31510  dvhgrp  31590  dvhlveclem  31591  dvh0g  31594  cdlemn8  31687  dihordlem7b  31698  dihopelvalcpre  31731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-fun 5415  df-fn 5416  df-f 5417
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