Theorem fco 3631

Description: Composition of two mappings.

Assertion

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

Proof of Theorem fco

Step	Hyp	Ref	Expression
1		funco 3546	. . . . . 6 |- ((Fun F /\ Fun G) -> Fun (F o. G))
2		ffun 3625	. . . . . 6 |- (F:B-->C -> Fun F)
3		ffun 3625	. . . . . 6 |- (G:A-->B -> Fun G)
4	1, 2, 3	syl2an 454	. . . . 5 |- ((F:B-->C /\ G:A-->B) -> Fun (F o. G))
5		fdm 3627	. . . . . . . . . 10 |- (F:B-->C -> dom F = B)
6	5	sseq2d 2086	. . . . . . . . 9 |- (F:B-->C -> (ran G (_ dom F <-> ran G (_ B))
7		frn 3628	. . . . . . . . 9 |- (G:A-->B -> ran G (_ B)
8	6, 7	syl5bir 210	. . . . . . . 8 |- (F:B-->C -> (G:A-->B -> ran G (_ dom F))
9	8	imp 350	. . . . . . 7 |- ((F:B-->C /\ G:A-->B) -> ran G (_ dom F)
10		dmcosseq 3361	. . . . . . 7 |- (ran G (_ dom F -> dom ( F o. G) = dom G)
11	9, 10	syl 10	. . . . . 6 |- ((F:B-->C /\ G:A-->B) -> dom ( F o. G) = dom G)
12		fdm 3627	. . . . . . 7 |- (G:A-->B -> dom G = A)
13	12	adantl 388	. . . . . 6 |- ((F:B-->C /\ G:A-->B) -> dom G = A)
14	11, 13	eqtrd 1505	. . . . 5 |- ((F:B-->C /\ G:A-->B) -> dom ( F o. G) = A)
15	4, 14	jca 288	. . . 4 |- ((F:B-->C /\ G:A-->B) -> (Fun (F o. G) /\ dom ( F o. G) = A))
16		df-fn 3189	. . . 4 |- ((F o. G) Fn A <-> (Fun (F o. G) /\ dom ( F o. G) = A))
17	15, 16	sylibr 200	. . 3 |- ((F:B-->C /\ G:A-->B) -> (F o. G) Fn A)
18		rncoss 3360	. . . . 5 |- ran ( F o. G) (_ ran F
19		sstr2 2068	. . . . . 6 |- (ran ( F o. G) (_ ran F -> (ran F (_ C -> ran ( F o. G) (_ C))
20		frn 3628	. . . . . 6 |- (F:B-->C -> ran F (_ C)
21	19, 20	syl5 21	. . . . 5 |- (ran ( F o. G) (_ ran F -> (F:B-->C -> ran ( F o. G) (_ C))
22	18, 21	ax-mp 7	. . . 4 |- (F:B-->C -> ran ( F o. G) (_ C)
23	22	adantr 389	. . 3 |- ((F:B-->C /\ G:A-->B) -> ran ( F o. G) (_ C)
24	17, 23	jca 288	. 2 |- ((F:B-->C /\ G:A-->B) -> ((F o. G) Fn A /\ ran ( F o. G) (_ C))
25		df-f 3190	. 2 |- ((F o. G):A-->C <-> ((F o. G) Fn A /\ ran ( F o. G) (_ C))
26	24, 25	sylibr 200	1 |- ((F:B-->C /\ G:A-->B) -> (F o. G):A-->C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   (_ wss 2044  dom cdm 3166  ran crn 3167   o. ccom 3170  Fun wfun 3172   Fn wfn 3173  -->wf 3174

This theorem is referenced by:  f1co 3662  foco 3677  mapenlem1 4478  mapenlem2 4479  ac6lem 4737  uzrdgfnuz 6256  ruclem17 7486  cnco 7728  cnpco 7729  cnmetba 7865  cnmet 7866  cncfmet 7867  remetba 7871  imsdf 8284  lnocoi 8380  sincolem 8619  hocof 9649  homco1t 9684  homco2t 9858  hmopcot 9904  pjinvar 10075

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-fun 3188  df-fn 3189  df-f 3190

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