Theorem fnfco 5511

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 5330	. 2 |- ( G : B --> A <-> ( G Fn B /\ ran G C_ A ) )
2		fnco 5440	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B )
3	2	3expb 1230	. 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 3200   ran crn 4726   o. ccom 4729   Fn wfn 5321   -->wf 5322

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-fun 5328  df-fn 5329  df-f 5330

This theorem is referenced by:  cocan1  5927  cocan2  5928  ofco  6253  1stcof  6325  2ndcof  6326  cc3  7486