Theorem fnco 5027

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

Assertion

Ref	Expression
fnco	|- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B )

Proof of Theorem fnco

Step	Hyp	Ref	Expression
1		fnfun 5016	. . . 4 |- ( F Fn A -> Fun F )
2		fnfun 5016	. . . 4 |- ( G Fn B -> Fun G )
3		funco 4960	. . . 4 |- ( ( Fun F /\ Fun G ) -> Fun ( F o. G ) )
4	1, 2, 3	syl2an 283	. . 3 |- ( ( F Fn A /\ G Fn B ) -> Fun ( F o. G ) )
5	4	3adant3 958	. 2 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> Fun ( F o. G ) )
6		fndm 5018	. . . . . . 7 |- ( F Fn A -> dom F = A )
7	6	sseq2d 3027	. . . . . 6 |- ( F Fn A -> ( ran G C_ dom F <-> ran G C_ A ) )
8	7	biimpar 291	. . . . 5 |- ( ( F Fn A /\ ran G C_ A ) -> ran G C_ dom F )
9		dmcosseq 4621	. . . . 5 |- ( ran G C_ dom F -> dom ( F o. G ) = dom G )
10	8, 9	syl 14	. . . 4 |- ( ( F Fn A /\ ran G C_ A ) -> dom ( F o. G ) = dom G )
11	10	3adant2 957	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom ( F o. G ) = dom G )
12		fndm 5018	. . . 4 |- ( G Fn B -> dom G = B )
13	12	3ad2ant2 960	. . 3 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom G = B )
14	11, 13	eqtrd 2113	. 2 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> dom ( F o. G ) = B )
15		df-fn 4925	. 2 |- ( ( F o. G ) Fn B <-> ( Fun ( F o. G ) /\ dom ( F o. G ) = B ) )
16	5, 14, 15	sylanbrc 408	1 |- ( ( F Fn A /\ G Fn B /\ ran G C_ A ) -> ( F o. G ) Fn B )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   C_ wss 2973   dom cdm 4363   ran crn 4364   o. ccom 4367   Fun wfun 4916   Fn wfn 4917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-fun 4924  df-fn 4925

This theorem is referenced by:  fco  5076  fnfco  5085