Theorem fnfco 5409

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 5239	. 2 |- ( G : B --> A <-> ( G Fn B /\ ran G C_ A ) )
2		fnco 5343	. . 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 3144   ran crn 4645   o. ccom 4648   Fn wfn 5230   -->wf 5231

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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-f 5239

This theorem is referenced by:  cocan1  5808  cocan2  5809  ofco  6124  1stcof  6187  2ndcof  6188  cc3  7296