Theorem fnco 3591

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 3546	. . . . 5 |- ((Fun F /\ Fun G) -> Fun (F o. G))
2		fnfun 3581	. . . . 5 |- (F Fn A -> Fun F)
3		fnfun 3581	. . . . 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 798	. . 3 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> Fun (F o. G))
6		fndm 3583	. . . . . . . 8 |- (F Fn A -> dom F = A)
7	6	sseq2d 2086	. . . . . . 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 3361	. . . . . 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 797	. . . 4 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> dom ( F o. G) = dom G)
12		fndm 3583	. . . . 5 |- (G Fn B -> dom G = B)
13	12	3ad2ant2 800	. . . 4 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> dom G = B)
14	11, 13	eqtrd 1505	. . 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 3189	. 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 774   = wceq 955   (_ wss 2044  dom cdm 3166  ran crn 3167   o. ccom 3170  Fun wfun 3172   Fn wfn 3173

This theorem is referenced by:  fnfco 3637  fopabco 3827  fopabcos 3828  0vfval 8189  cayleylem2 10366

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-3an 776  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

Copyright terms: Public domain