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Theorem fcoi1 5303
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 5272 . 2  |-  ( F : A --> B  ->  F  Fn  A )
2 df-fn 5126 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
3 eqimss 3151 . . . . 5  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
4 cnvi 4943 . . . . . . . . . 10  |-  `'  _I  =  _I
54reseq1i 4815 . . . . . . . . 9  |-  ( `'  _I  |`  A )  =  (  _I  |`  A )
65cnveqi 4714 . . . . . . . 8  |-  `' ( `'  _I  |`  A )  =  `' (  _I  |`  A )
7 cnvresid 5197 . . . . . . . 8  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
86, 7eqtr2i 2161 . . . . . . 7  |-  (  _I  |`  A )  =  `' ( `'  _I  |`  A )
98coeq2i 4699 . . . . . 6  |-  ( F  o.  (  _I  |`  A ) )  =  ( F  o.  `' ( `'  _I  |`  A )
)
10 cores2 5051 . . . . . 6  |-  ( dom 
F  C_  A  ->  ( F  o.  `' ( `'  _I  |`  A )
)  =  ( F  o.  _I  ) )
119, 10syl5eq 2184 . . . . 5  |-  ( dom 
F  C_  A  ->  ( F  o.  (  _I  |`  A ) )  =  ( F  o.  _I  ) )
123, 11syl 14 . . . 4  |-  ( dom 
F  =  A  -> 
( F  o.  (  _I  |`  A ) )  =  ( F  o.  _I  ) )
13 funrel 5140 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
14 coi1 5054 . . . . 5  |-  ( Rel 
F  ->  ( F  o.  _I  )  =  F )
1513, 14syl 14 . . . 4  |-  ( Fun 
F  ->  ( F  o.  _I  )  =  F )
1612, 15sylan9eqr 2194 . . 3  |-  ( ( Fun  F  /\  dom  F  =  A )  -> 
( F  o.  (  _I  |`  A ) )  =  F )
172, 16sylbi 120 . 2  |-  ( F  Fn  A  ->  ( F  o.  (  _I  |`  A ) )  =  F )
181, 17syl 14 1  |-  ( F : A --> B  -> 
( F  o.  (  _I  |`  A ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    C_ wss 3071    _I cid 4210   `'ccnv 4538   dom cdm 4539    |` cres 4541    o. ccom 4543   Rel wrel 4544   Fun wfun 5117    Fn wfn 5118   -->wf 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fun 5125  df-fn 5126  df-f 5127
This theorem is referenced by:  fcof1o  5690  mapen  6740  hashfacen  10593
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