Theorem fco2 5529

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

Step	Hyp	Ref	Expression
1		fco 5527	. 2 |- ( ( ( F |` B ) : B --> C /\ G : A --> B ) -> ( ( F |` B ) o. G ) : A --> C )
2		frn 5517	. . . . 5 |- ( G : A --> B -> ran G C_ B )
3		cores 5266	. . . . 5 |- ( ran G C_ B -> ( ( F |` B ) o. G ) = ( F o. G ) )
4	2, 3	syl 14	. . . 4 |- ( G : A --> B -> ( ( F |` B ) o. G ) = ( F o. G ) )
5	4	adantl 277	. . 3 |- ( ( ( F |` B ) : B --> C /\ G : A --> B ) -> ( ( F |` B ) o. G ) = ( F o. G ) )
6	5	feq1d 5495	. 2 |- ( ( ( F |` B ) : B --> C /\ G : A --> B ) -> ( ( ( F |` B ) o. G ) : A --> C <-> ( F o. G ) : A --> C ) )
7	1, 6	mpbid 147	1 |- ( ( ( F |` B ) : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   C_ wss 3211   ran crn 4750   |` cres 4751   o. ccom 4753   -->wf 5348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-fun 5354  df-fn 5355  df-f 5356

This theorem is referenced by:  isomninnlem  16814  iswomninnlem  16834  ismkvnnlem  16837