Theorem fnfco 5428

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 5258	. 2 |- ( G : B --> A <-> ( G Fn B /\ ran G C_ A ) )
2		fnco 5362	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B )
3	2	3expb 1206	. 2 |- ( ( F Fn A /\ ( G Fn B /\ ran G C_ A ) ) -> ( F o. G ) Fn B )
4	1, 3	sylan2b 287	1 |- ( ( F Fn A /\ G : B --> A ) -> ( F o. G ) Fn B )

Syntax hints:   -> wi 4   /\ wa 104   C_ wss 3153   ran crn 4660   o. ccom 4663   Fn wfn 5249   -->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-fun 5256  df-fn 5257  df-f 5258

This theorem is referenced by:  cocan1  5830  cocan2  5831  ofco  6149  1stcof  6216  2ndcof  6217  cc3  7328