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Theorem fcoi2 5522
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 5362 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 cores 5279 . . 3  |-  ( ran 
F  C_  B  ->  ( (  _I  |`  B )  o.  F )  =  (  _I  o.  F
) )
3 fnrel 5447 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
4 coi2 5292 . . . 4  |-  ( Rel 
F  ->  (  _I  o.  F )  =  F )
53, 4syl 15 . . 3  |-  ( F  Fn  A  ->  (  _I  o.  F )  =  F )
62, 5sylan9eqr 2420 . 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 1647    C_ wss 3238    _I cid 4407   ran crn 4793    |` cres 4794    o. ccom 4796   Rel wrel 4797    Fn wfn 5353   -->wf 5354
This theorem is referenced by:  fcof1o  5926  mapen  7168  mapfien  7546  hashfacen  11590  cofulid  13974  setccatid  14126  symggrp  14990  gsumval3  15401  gsumzf1o  15406  frgpcyg  16744  qtophmeo  17725  hoico2  22771  subfacp1lem5  24439  f1linds  26886  f1omvdco2  26982  symggen  27002  psgnunilem1  27007  mendrng  27091  ltrncoidN  30376  trlcoat  30971  trlcone  30976  cdlemg47a  30982  cdlemg47  30984  trljco  30988  tgrpgrplem  30997  tendo1mul  31018  tendo0pl  31039  cdlemkid2  31172  cdlemk45  31195  cdlemk53b  31204  erng1r  31243  tendocnv  31270  dvalveclem  31274  dva0g  31276  dvhgrp  31356  dvhlveclem  31357  dvh0g  31360  cdlemn8  31453  dihordlem7b  31464  dihopelvalcpre  31497
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-fun 5360  df-fn 5361  df-f 5362
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