Theorem fco2 5549

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 5547	. 2 |- ( ( ( F |` B ) : B --> C /\ G : A --> B ) -> ( ( F |` B ) o. G ) : A --> C )
2		frn 5537	. . . . 5 |- ( G : A --> B -> ran G C_ B )
3		cores 5286	. . . . 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 5515	. 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 1402   C_ wss 3220   ran crn 4770   |` cres 4771   o. ccom 4773   -->wf 5368

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-fun 5374  df-fn 5375  df-f 5376

This theorem is referenced by:  isomninnlem  16984  iswomninnlem  17004  ismkvnnlem  17007