Theorem fnun 3590

Description: The union of two functions with disjoint domains.

Assertion

Ref	Expression
fnun	|- (((F Fn A /\ G Fn B) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B))

Proof of Theorem fnun

Step	Hyp	Ref	Expression
1		ineq12 2209	. . . . . . . . . . 11 |- ((dom F = A /\ dom G = B) -> (dom F i^i dom G) = (A i^i B))
2	1	eqeq1d 1481	. . . . . . . . . 10 |- ((dom F = A /\ dom G = B) -> ((dom F i^i dom G) = (/) <-> (A i^i B) = (/)))
3	2	anbi2d 615	. . . . . . . . 9 |- ((dom F = A /\ dom G = B) -> (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) <-> ((Fun F /\ Fun G) /\ (A i^i B) = (/))))
4		funun 3550	. . . . . . . . 9 |- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> Fun (F u. G))
5	3, 4	syl6bir 215	. . . . . . . 8 |- ((dom F = A /\ dom G = B) -> (((Fun F /\ Fun G) /\ (A i^i B) = (/)) -> Fun (F u. G)))
6		uneq12 2176	. . . . . . . . 9 |- ((dom F = A /\ dom G = B) -> (dom F u. dom G) = (A u. B))
7		dmun 3313	. . . . . . . . 9 |- dom ( F u. G) = (dom F u. dom G)
8	6, 7	syl5eq 1517	. . . . . . . 8 |- ((dom F = A /\ dom G = B) -> dom ( F u. G) = (A u. B))
9	5, 8	jctird 601	. . . . . . 7 |- ((dom F = A /\ dom G = B) -> (((Fun F /\ Fun G) /\ (A i^i B) = (/)) -> (Fun (F u. G) /\ dom ( F u. G) = (A u. B))))
10		df-fn 3189	. . . . . . 7 |- ((F u. G) Fn (A u. B) <-> (Fun (F u. G) /\ dom ( F u. G) = (A u. B)))
11	9, 10	syl6ibr 213	. . . . . 6 |- ((dom F = A /\ dom G = B) -> (((Fun F /\ Fun G) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B)))
12	11	exp3a 375	. . . . 5 |- ((dom F = A /\ dom G = B) -> ((Fun F /\ Fun G) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B))))
13	12	impcom 351	. . . 4 |- (((Fun F /\ Fun G) /\ (dom F = A /\ dom G = B)) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B)))
14	13	an4s 508	. . 3 |- (((Fun F /\ dom F = A) /\ (Fun G /\ dom G = B)) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B)))
15		df-fn 3189	. . 3 |- (F Fn A <-> (Fun F /\ dom F = A))
16		df-fn 3189	. . 3 |- (G Fn B <-> (Fun G /\ dom G = B))
17	14, 15, 16	syl2anb 455	. 2 |- ((F Fn A /\ G Fn B) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B)))
18	17	imp 350	1 |- (((F Fn A /\ G Fn B) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   u. cun 2042   i^i cin 2043  (/)c0 2277  dom cdm 3166  Fun wfun 3172   Fn wfn 3173

This theorem is referenced by:  fun 3636  f1oun 3701

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-ral 1647  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-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-fun 3188  df-fn 3189

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