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Theorem fco2 5603
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 5602 . 2  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( F  |`  B )  o.  G
) : A --> C )
2 frn 5599 . . . . 5  |-  ( G : A --> B  ->  ran  G  C_  B )
3 cores 5375 . . . . 5  |-  ( ran 
G  C_  B  ->  ( ( F  |`  B )  o.  G )  =  ( F  o.  G
) )
42, 3syl 16 . . . 4  |-  ( G : A --> B  -> 
( ( F  |`  B )  o.  G
)  =  ( F  o.  G ) )
54adantl 454 . . 3  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( F  |`  B )  o.  G
)  =  ( F  o.  G ) )
65feq1d 5582 . 2  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( ( F  |`  B )  o.  G ) : A --> C 
<->  ( F  o.  G
) : A --> C ) )
71, 6mpbid 203 1  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    C_ wss 3322   ran crn 4881    |` cres 4882    o. ccom 4884   -->wf 5452
This theorem is referenced by:  prdsrngd  15720  prdscrngd  15721  prds1  15722  prdstmdd  18155  prdsxmslem2  18561  ftc1anclem3  26284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-fun 5458  df-fn 5459  df-f 5460
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