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Theorem fco2 5747
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 5746 . 2  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( F  |`  B )  o.  G
) : A --> C )
2 frn 5742 . . . . 5  |-  ( G : A --> B  ->  ran  G  C_  B )
3 cores 5515 . . . . 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 466 . . 3  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( F  |`  B )  o.  G
)  =  ( F  o.  G ) )
65feq1d 5722 . 2  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( ( ( F  |`  B )  o.  G ) : A --> C 
<->  ( F  o.  G
) : A --> C ) )
71, 6mpbid 210 1  |-  ( ( ( F  |`  B ) : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    C_ wss 3481   ran crn 5005    |` cres 5006    o. ccom 5008   -->wf 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-fun 5595  df-fn 5596  df-f 5597
This theorem is referenced by:  fsuppcor  7873  prdsringd  17110  prdscrngd  17111  prds1  17112  prdstmdd  20467  prdsxmslem2  20877  eulerpartlemmf  28107  sseqf  28124  ftc1anclem3  29987
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