Theorem fco2 5374

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 5373	. 2 |- ( ( ( F |` B ) : B --> C /\ G : A --> B ) -> ( ( F |` B ) o. G ) : A --> C )
2		frn 5366	. . . . 5 |- ( G : A --> B -> ran G C_ B )
3		cores 5124	. . . . 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 5344	. 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 1353   C_ wss 3127   ran crn 4621   |` cres 4622   o. ccom 4624   -->wf 5204

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-fun 5210  df-fn 5211  df-f 5212

This theorem is referenced by:  isomninnlem  14337  iswomninnlem  14356  ismkvnnlem  14359