Theorem fnco 3595

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 3550	. . . . 5 |- ((Fun F /\ Fun G) -> Fun (F o. G))
2		fnfun 3585	. . . . 5 |- (F Fn A -> Fun F)
3		fnfun 3585	. . . . 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 799	. . 3 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> Fun (F o. G))
6		fndm 3587	. . . . . . . 8 |- (F Fn A -> dom F = A)
7	6	sseq2d 2089	. . . . . . 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 3365	. . . . . 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 798	. . . 4 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> dom ( F o. G) = dom G)
12		fndm 3587	. . . . 5 |- (G Fn B -> dom G = B)
13	12	3ad2ant2 801	. . . 4 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> dom G = B)
14	11, 13	eqtrd 1507	. . 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 3193	. 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 775   = wceq 956   (_ wss 2047  dom cdm 3170  ran crn 3171   o. ccom 3174  Fun wfun 3176   Fn wfn 3177

This theorem is referenced by:  fnfco 3642  fopabco 3832  fopabcos 3833  0vfval 8225  cayleylem2 10410

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193

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