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Theorem fcoi1 5667
Description: Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fcoi1  |-  ( F : A --> B  -> 
( F  o.  (  _I  |`  A ) )  =  F )

Proof of Theorem fcoi1
StepHypRef Expression
1 ffn 5639 . 2  |-  ( F : A --> B  ->  F  Fn  A )
2 df-fn 5499 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
3 eqimss 3469 . . . . 5  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
4 cnvi 5320 . . . . . . . . . 10  |-  `'  _I  =  _I
54reseq1i 5182 . . . . . . . . 9  |-  ( `'  _I  |`  A )  =  (  _I  |`  A )
65cnveqi 5090 . . . . . . . 8  |-  `' ( `'  _I  |`  A )  =  `' (  _I  |`  A )
7 cnvresid 5566 . . . . . . . 8  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
86, 7eqtr2i 2412 . . . . . . 7  |-  (  _I  |`  A )  =  `' ( `'  _I  |`  A )
98coeq2i 5076 . . . . . 6  |-  ( F  o.  (  _I  |`  A ) )  =  ( F  o.  `' ( `'  _I  |`  A )
)
10 cores2 5428 . . . . . 6  |-  ( dom 
F  C_  A  ->  ( F  o.  `' ( `'  _I  |`  A )
)  =  ( F  o.  _I  ) )
119, 10syl5eq 2435 . . . . 5  |-  ( dom 
F  C_  A  ->  ( F  o.  (  _I  |`  A ) )  =  ( F  o.  _I  ) )
123, 11syl 16 . . . 4  |-  ( dom 
F  =  A  -> 
( F  o.  (  _I  |`  A ) )  =  ( F  o.  _I  ) )
13 funrel 5513 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
14 coi1 5431 . . . . 5  |-  ( Rel 
F  ->  ( F  o.  _I  )  =  F )
1513, 14syl 16 . . . 4  |-  ( Fun 
F  ->  ( F  o.  _I  )  =  F )
1612, 15sylan9eqr 2445 . . 3  |-  ( ( Fun  F  /\  dom  F  =  A )  -> 
( F  o.  (  _I  |`  A ) )  =  F )
172, 16sylbi 195 . 2  |-  ( F  Fn  A  ->  ( F  o.  (  _I  |`  A ) )  =  F )
181, 17syl 16 1  |-  ( F : A --> B  -> 
( F  o.  (  _I  |`  A ) )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    C_ wss 3389    _I cid 4704   `'ccnv 4912   dom cdm 4913    |` cres 4915    o. ccom 4917   Rel wrel 4918   Fun wfun 5490    Fn wfn 5491   -->wf 5492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-fun 5498  df-fn 5499  df-f 5500
This theorem is referenced by:  fcof1oinvd  6097  mapen  7600  mapfien  7782  mapfienOLD  8051  hashfacen  12407  cofurid  15297  setccatid  15480  estrccatid  15518  curf2ndf  15633  symgid  16543  f1omvdco2  16590  psgnunilem1  16635  pf1mpf  18501  pf1ind  18504  wilthlem3  23461  hoico1  26791  fcobijfs  27699  diophrw  30857  mendring  31309  rngccatidALTV  32997  ringccatidALTV  33060  ltrncoidN  36265  trlcoabs2N  36861  trlcoat  36862  cdlemg47a  36873  cdlemg46  36874  trljco  36879  tendo1mulr  36910  tendo0co2  36927  cdlemi2  36958  cdlemk2  36971  cdlemk4  36973  cdlemk8  36977  cdlemk53  37096  cdlemk55a  37098  dvhopN  37256  dihopelvalcpre  37388  dihmeetlem1N  37430  dihglblem5apreN  37431
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