Theorem fnfco 5362

Description: Composition of two functions. (Contributed by NM, 22-May-2006.)

Assertion

Ref	Expression
fnfco	|- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B )

Proof of Theorem fnfco

Step	Hyp	Ref	Expression
1		df-f 5192	. 2 |- ( G : B --> A <-> ( G Fn B /\ ran G C_ A ) )
2		fnco 5296	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B )
3	2	3expb 1194	. 2 |- ( ( F Fn A /\ ( G Fn B /\ ran G C_ A ) ) -> ( F o. G ) Fn B )
4	1, 3	sylan2b 285	1 |- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3116   ran crn 4605   o. ccom 4608   Fn wfn 5183   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  cocan1  5755  cocan2  5756  ofco  6068  1stcof  6131  2ndcof  6132  cc3  7209