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Theorem fco2 5259
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 5258 . 2  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( F  |`  B )  o.  G
) : A --> C )
2 frn 5251 . . . . 5  |-  ( G : A --> B  ->  ran  G  C_  B )
3 cores 5012 . . . . 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 5229 . 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 1316    C_ wss 3041   ran crn 4510    |` cres 4511    o. ccom 4513   -->wf 5089
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-fun 5095  df-fn 5096  df-f 5097
This theorem is referenced by:  isomninnlem  13152
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