Theorem fcoOLD 4574

Description: Composition of two mappings.

Assertion

Ref	Expression
fcoOLD	|- ((F:B-->C /\ G:A-->B) -> (F o. G):A-->C)

Proof of Theorem fcoOLD

Step	Hyp	Ref	Expression
1		df-f 4010	. 2 |- ((F o. G):A-->C <-> ((F o. G) Fn A /\ ran ( F o. G) C_ C))
2		df-fn 4009	. . 3 |- ((F o. G) Fn A <-> (Fun (F o. G) /\ dom ( F o. G) = A))
3		funco 4457	. . . 4 |- ((Fun F /\ Fun G) -> Fun (F o. G))
4		ffun 4565	. . . 4 |- (F:B-->C -> Fun F)
5		ffun 4565	. . . 4 |- (G:A-->B -> Fun G)
6	3, 4, 5	syl2an 503	. . 3 |- ((F:B-->C /\ G:A-->B) -> Fun (F o. G))
7		fdm 4567	. . . . . . . 8 |- (F:B-->C -> dom F = B)
8	7	sseq2d 2645	. . . . . . 7 |- (F:B-->C -> (ran G C_ dom F <-> ran G C_ B))
9		frn 4569	. . . . . . 7 |- (G:A-->B -> ran G C_ B)
10	8, 9	syl5bir 227	. . . . . 6 |- (F:B-->C -> (G:A-->B -> ran G C_ dom F))
11	10	imp 377	. . . . 5 |- ((F:B-->C /\ G:A-->B) -> ran G C_ dom F)
12		dmcosseq 4214	. . . . 5 |- (ran G C_ dom F -> dom ( F o. G) = dom G)
13	11, 12	syl 12	. . . 4 |- ((F:B-->C /\ G:A-->B) -> dom ( F o. G) = dom G)
14		fdm 4567	. . . . 5 |- (G:A-->B -> dom G = A)
15	14	adantl 424	. . . 4 |- ((F:B-->C /\ G:A-->B) -> dom G = A)
16	13, 15	eqtrd 1925	. . 3 |- ((F:B-->C /\ G:A-->B) -> dom ( F o. G) = A)
17	2, 6, 16	sylanbrc 527	. 2 |- ((F:B-->C /\ G:A-->B) -> (F o. G) Fn A)
18		rncoss 4213	. . . 4 |- ran ( F o. G) C_ ran F
19		sstr2 2623	. . . . 5 |- (ran ( F o. G) C_ ran F -> (ran F C_ C -> ran ( F o. G) C_ C))
20		frn 4569	. . . . 5 |- (F:B-->C -> ran F C_ C)
21	19, 20	syl5 20	. . . 4 |- (ran ( F o. G) C_ ran F -> (F:B-->C -> ran ( F o. G) C_ C))
22	18, 21	ax-mp 7	. . 3 |- (F:B-->C -> ran ( F o. G) C_ C)
23	22	adantr 425	. 2 |- ((F:B-->C /\ G:A-->B) -> ran ( F o. G) C_ C)
24	1, 17, 23	sylanbrc 527	1 |- ((F:B-->C /\ G:A-->B) -> (F o. G):A-->C)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   C_ wss 2593  dom cdm 3986  ran crn 3987   o. ccom 3990  Fun wfun 3992   Fn wfn 3993  -->wf 3994

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010

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