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Theorem fco2 5415
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 5414 . 2  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( F  |`  B )  o.  G
) : A --> C )
2 frn 5411 . . . . 5  |-  ( G : A --> B  ->  ran  G  C_  B )
3 cores 5192 . . . . 5  |-  ( ran 
G  C_  B  ->  ( ( F  |`  B )  o.  G )  =  ( F  o.  G
) )
42, 3syl 15 . . . 4  |-  ( G : A --> B  -> 
( ( F  |`  B )  o.  G
)  =  ( F  o.  G ) )
54adantl 452 . . 3  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( F  |`  B )  o.  G
)  =  ( F  o.  G ) )
65feq1d 5395 . 2  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( ( F  |`  B )  o.  G ) : A --> C 
<->  ( F  o.  G
) : A --> C ) )
71, 6mpbid 201 1  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    C_ wss 3165   ran crn 4706    |` cres 4707    o. ccom 4709   -->wf 5267
This theorem is referenced by:  prdsrngd  15411  prdscrngd  15412  prds1  15413  prdstmdd  17822  prdsxmslem2  18091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-fun 5273  df-fn 5274  df-f 5275
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