Theorem fnco 3581

Description: Composition of two functions.

Assertion

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

Proof of Theorem fnco

Step	Hyp	Ref	Expression
1		funco 3536	. . . . 5 |- ((Fun F /\ Fun G) -> Fun (F o. G))
2		fnfun 3571	. . . . 5 |- (F Fn A -> Fun F)
3		fnfun 3571	. . . . 5 |- (G Fn B -> Fun G)
4	1, 2, 3	syl2an 454	. . . 4 |- ((F Fn A /\ G Fn B) -> Fun (F o. G))
5	4	3adant3 797	. . 3 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> Fun (F o. G))
6		fndm 3573	. . . . . . . 8 |- (F Fn A -> dom F = A)
7	6	sseq2d 2079	. . . . . . 7 |- (F Fn A -> (ran G (_ dom F <-> ran G (_ A))
8	7	biimpar 417	. . . . . 6 |- ((F Fn A /\ ran G (_ A) -> ran G (_ dom F)
9		dmcosseq 3349	. . . . . 6 |- (ran G (_ dom F -> dom ( F o. G) = dom G)
10	8, 9	syl 10	. . . . 5 |- ((F Fn A /\ ran G (_ A) -> dom ( F o. G) = dom G)
11	10	3adant2 796	. . . 4 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> dom ( F o. G) = dom G)
12		fndm 3573	. . . . 5 |- (G Fn B -> dom G = B)
13	12	3ad2ant2 799	. . . 4 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> dom G = B)
14	11, 13	eqtrd 1499	. . 3 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> dom ( F o. G) = B)
15	5, 14	jca 288	. 2 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> (Fun (F o. G) /\ dom ( F o. G) = B))
16		df-fn 3183	. 2 |- ((F o. G) Fn B <-> (Fun (F o. G) /\ dom ( F o. G) = B))
17	15, 16	sylibr 200	1 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> (F o. G) Fn B)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   (_ wss 2037  dom cdm 3160  ran crn 3161   o. ccom 3164  Fun wfun 3166   Fn wfn 3167

This theorem is referenced by:  fnfco 3627  fopabco 3817  fopabcos 3818  0vfval 8163  cayleylem2 10317

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-fun 3182  df-fn 3183

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