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Theorem fco2 5289
Description: Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
Assertion
Ref Expression
fco2  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )

Proof of Theorem fco2
StepHypRef Expression
1 fco 5288 . 2  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( F  |`  B )  o.  G
) : A --> C )
2 frn 5281 . . . . 5  |-  ( G : A --> B  ->  ran  G  C_  B )
3 cores 5042 . . . . 5  |-  ( ran 
G  C_  B  ->  ( ( F  |`  B )  o.  G )  =  ( F  o.  G
) )
42, 3syl 14 . . . 4  |-  ( G : A --> B  -> 
( ( F  |`  B )  o.  G
)  =  ( F  o.  G ) )
54adantl 275 . . 3  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( F  |`  B )  o.  G
)  =  ( F  o.  G ) )
65feq1d 5259 . 2  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( ( F  |`  B )  o.  G ) : A --> C 
<->  ( F  o.  G
) : A --> C ) )
71, 6mpbid 146 1  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    C_ wss 3071   ran crn 4540    |` cres 4541    o. ccom 4543   -->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-fun 5125  df-fn 5126  df-f 5127
This theorem is referenced by:  isomninnlem  13255
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