Theorem fco2 5420

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 5419	. 2 |- ( ( ( F |` B ) : B --> C /\ G : A --> B ) -> ( ( F |` B ) o. G ) : A --> C )
2		frn 5412	. . . . 5 |- ( G : A --> B -> ran G C_ B )
3		cores 5169	. . . . 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 5390	. 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 1364   C_ wss 3153   ran crn 4660   |` cres 4661   o. ccom 4663   -->wf 5250

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-fun 5256  df-fn 5257  df-f 5258

This theorem is referenced by:  isomninnlem  15520  iswomninnlem  15539  ismkvnnlem  15542