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Theorem fcoi1 5550
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 5524 . 2  |-  ( F : A --> B  ->  F  Fn  A )
2 df-fn 5390 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
3 eqimss 3336 . . . . 5  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
4 cnvi 5209 . . . . . . . . . 10  |-  `'  _I  =  _I
54reseq1i 5075 . . . . . . . . 9  |-  ( `'  _I  |`  A )  =  (  _I  |`  A )
65cnveqi 4980 . . . . . . . 8  |-  `' ( `'  _I  |`  A )  =  `' (  _I  |`  A )
7 cnvresid 5456 . . . . . . . 8  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
86, 7eqtr2i 2401 . . . . . . 7  |-  (  _I  |`  A )  =  `' ( `'  _I  |`  A )
98coeq2i 4966 . . . . . 6  |-  ( F  o.  (  _I  |`  A ) )  =  ( F  o.  `' ( `'  _I  |`  A )
)
10 cores2 5315 . . . . . 6  |-  ( dom 
F  C_  A  ->  ( F  o.  `' ( `'  _I  |`  A )
)  =  ( F  o.  _I  ) )
119, 10syl5eq 2424 . . . . 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 5404 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
14 coi1 5318 . . . . 5  |-  ( Rel 
F  ->  ( F  o.  _I  )  =  F )
1513, 14syl 16 . . . 4  |-  ( Fun 
F  ->  ( F  o.  _I  )  =  F )
1612, 15sylan9eqr 2434 . . 3  |-  ( ( Fun  F  /\  dom  F  =  A )  -> 
( F  o.  (  _I  |`  A ) )  =  F )
172, 16sylbi 188 . 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    C_ wss 3256    _I cid 4427   `'ccnv 4810   dom cdm 4811    |` cres 4813    o. ccom 4815   Rel wrel 4816   Fun wfun 5381    Fn wfn 5382   -->wf 5383
This theorem is referenced by:  fcof1o  5958  mapen  7200  mapfien  7579  hashfacen  11623  cofurid  14008  setccatid  14159  curf2ndf  14264  symgid  15024  pf1mpf  19832  pf1ind  19835  wilthlem3  20713  hoico1  23100  diophrw  26501  f1omvdco2  27053  psgnunilem1  27078  mendrng  27162  ltrncoidN  30293  trlcoabs2N  30887  trlcoat  30888  cdlemg47a  30899  cdlemg46  30900  trljco  30905  tendo1mulr  30936  tendo0co2  30953  cdlemi2  30984  cdlemk2  30997  cdlemk4  30999  cdlemk8  31003  cdlemk53  31122  cdlemk55a  31124  dvhopN  31282  dihopelvalcpre  31414  dihmeetlem1N  31456  dihglblem5apreN  31457
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-fun 5389  df-fn 5390  df-f 5391
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