Theorem fnco 4521

Description: Composition of two functions.

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		df-fn 4009	. 2 |- ((F o. G) Fn B <-> (Fun (F o. G) /\ dom ( F o. G) = B))
2		funco 4457	. . . 4 |- ((Fun F /\ Fun G) -> Fun (F o. G))
3		fnfun 4510	. . . 4 |- (F Fn A -> Fun F)
4		fnfun 4510	. . . 4 |- (G Fn B -> Fun G)
5	2, 3, 4	syl2an 503	. . 3 |- ((F Fn A /\ G Fn B) -> Fun (F o. G))
6	5	3adant3 896	. 2 |- ((F Fn A /\ G Fn B /\ ran G C_ A) -> Fun (F o. G))
7		fndm 4512	. . . . . . 7 |- (F Fn A -> dom F = A)
8	7	sseq2d 2645	. . . . . 6 |- (F Fn A -> (ran G C_ dom F <-> ran G C_ A))
9	8	biimpar 461	. . . . 5 |- ((F Fn A /\ ran G C_ A) -> ran G C_ dom F)
10		dmcosseq 4214	. . . . 5 |- (ran G C_ dom F -> dom ( F o. G) = dom G)
11	9, 10	syl 12	. . . 4 |- ((F Fn A /\ ran G C_ A) -> dom ( F o. G) = dom G)
12	11	3adant2 895	. . 3 |- ((F Fn A /\ G Fn B /\ ran G C_ A) -> dom ( F o. G) = dom G)
13		fndm 4512	. . . 4 |- (G Fn B -> dom G = B)
14	13	3ad2ant2 898	. . 3 |- ((F Fn A /\ G Fn B /\ ran G C_ A) -> dom G = B)
15	12, 14	eqtrd 1925	. 2 |- ((F Fn A /\ G Fn B /\ ran G C_ A) -> dom ( F o. G) = B)
16	1, 6, 15	sylanbrc 527	1 |- ((F Fn A /\ G Fn B /\ ran G C_ A) -> (F o. G) Fn B)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   C_ wss 2593  dom cdm 3986  ran crn 3987   o. ccom 3990  Fun wfun 3992   Fn wfn 3993

This theorem is referenced by:  fco 4573  fnfco 4581  fopabco 4805  fopabcos 4806  0vfval 9557  oprabco 10159  upxp 10225  uptx 10226  txcnopab 10228  cayleylem2 13642  fnopabco2b 14734  fnopabco 15711  upixp 15729  heiborlem33 15987  phtpycolem4 16054  pcocn 16076

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-3an 860  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

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