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Theorem fcoi1 5101
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 5077 . 2  |-  ( F : A --> B  ->  F  Fn  A )
2 df-fn 4935 . . 3  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
3 eqimss 3052 . . . . 5  |-  ( dom 
F  =  A  ->  dom  F  C_  A )
4 cnvi 4758 . . . . . . . . . 10  |-  `'  _I  =  _I
54reseq1i 4636 . . . . . . . . 9  |-  ( `'  _I  |`  A )  =  (  _I  |`  A )
65cnveqi 4538 . . . . . . . 8  |-  `' ( `'  _I  |`  A )  =  `' (  _I  |`  A )
7 cnvresid 5004 . . . . . . . 8  |-  `' (  _I  |`  A )  =  (  _I  |`  A )
86, 7eqtr2i 2103 . . . . . . 7  |-  (  _I  |`  A )  =  `' ( `'  _I  |`  A )
98coeq2i 4524 . . . . . 6  |-  ( F  o.  (  _I  |`  A ) )  =  ( F  o.  `' ( `'  _I  |`  A )
)
10 cores2 4863 . . . . . 6  |-  ( dom 
F  C_  A  ->  ( F  o.  `' ( `'  _I  |`  A )
)  =  ( F  o.  _I  ) )
119, 10syl5eq 2126 . . . . 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 4949 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
14 coi1 4866 . . . . 5  |-  ( Rel 
F  ->  ( F  o.  _I  )  =  F )
1513, 14syl 14 . . . 4  |-  ( Fun 
F  ->  ( F  o.  _I  )  =  F )
1612, 15sylan9eqr 2136 . . 3  |-  ( ( Fun  F  /\  dom  F  =  A )  -> 
( F  o.  (  _I  |`  A ) )  =  F )
172, 16sylbi 119 . 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 102    = wceq 1285    C_ wss 2974    _I cid 4051   `'ccnv 4370   dom cdm 4371    |` cres 4373    o. ccom 4375   Rel wrel 4376   Fun wfun 4926    Fn wfn 4927   -->wf 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-fun 4934  df-fn 4935  df-f 4936
This theorem is referenced by:  fcof1o  5460
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