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Theorem fcoi2 5585
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 5425 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 cores 5340 . . 3  |-  ( ran 
F  C_  B  ->  ( (  _I  |`  B )  o.  F )  =  (  _I  o.  F
) )
3 fnrel 5510 . . . 4  |-  ( F  Fn  A  ->  Rel  F )
4 coi2 5353 . . . 4  |-  ( Rel 
F  ->  (  _I  o.  F )  =  F )
53, 4syl 16 . . 3  |-  ( F  Fn  A  ->  (  _I  o.  F )  =  F )
62, 5sylan9eqr 2466 . 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 3288    _I cid 4461   ran crn 4846    |` cres 4847    o. ccom 4849   Rel wrel 4850    Fn wfn 5416   -->wf 5417
This theorem is referenced by:  fcof1o  5993  mapen  7238  mapfien  7617  hashfacen  11666  cofulid  14050  setccatid  14202  symggrp  15066  gsumval3  15477  gsumzf1o  15482  frgpcyg  16817  qtophmeo  17810  hoico2  23221  subfacp1lem5  24831  f1linds  27171  f1omvdco2  27267  symggen  27287  psgnunilem1  27292  mendrng  27376  ltrncoidN  30622  trlcoat  31217  trlcone  31222  cdlemg47a  31228  cdlemg47  31230  trljco  31234  tgrpgrplem  31243  tendo1mul  31264  tendo0pl  31285  cdlemkid2  31418  cdlemk45  31441  cdlemk53b  31450  erng1r  31489  tendocnv  31516  dvalveclem  31520  dva0g  31522  dvhgrp  31602  dvhlveclem  31603  dvh0g  31606  cdlemn8  31699  dihordlem7b  31710  dihopelvalcpre  31743
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 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-fun 5423  df-fn 5424  df-f 5425
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