Theorem fnun 2730

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 1640	. . . . . . . . . . . 12 |- ((dom F = A /\ dom G = B) -> (dom F i^i dom G) = (A i^i B))
2	1	cleq1d 1109	. . . . . . . . . . 11 |- ((dom F = A /\ dom G = B) -> ((dom F i^i dom G) = (/) <-> (A i^i B) = (/)))
3	2	anbi2d 468	. . . . . . . . . 10 |- ((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 2700	. . . . . . . . . 10 |- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> Fun (F u. G))
5	3, 4	syl6bir 188	. . . . . . . . 9 |- ((dom F = A /\ dom G = B) -> (((Fun F /\ Fun G) /\ (A i^i B) = (/)) -> Fun (F u. G)))
6		uneq12 1613	. . . . . . . . . . 11 |- ((dom F = A /\ dom G = B) -> (dom F u. dom G) = (A u. B))
7		dmun 2536	. . . . . . . . . . 11 |- dom (F u. G) = (dom F u. dom G)
8	6, 7	syl5eq 1136	. . . . . . . . . 10 |- ((dom F = A /\ dom G = B) -> dom (F u. G) = (A u. B))
9	8	a1d 14	. . . . . . . . 9 |- ((dom F = A /\ dom G = B) -> (((Fun F /\ Fun G) /\ (A i^i B) = (/)) -> dom (F u. G) = (A u. B)))
10	5, 9	jcad 455	. . . . . . . 8 |- ((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))))
11		df-fn 2433	. . . . . . . 8 |- ((F u. G) Fn (A u. B) <-> (Fun (F u. G) /\ dom (F u. G) = (A u. B)))
12	10, 11	syl6ibr 186	. . . . . . 7 |- ((dom F = A /\ dom G = B) -> (((Fun F /\ Fun G) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B)))
13	12	exp3a 292	. . . . . 6 |- ((dom F = A /\ dom G = B) -> ((Fun F /\ Fun G) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B))))
14	13	com12 13	. . . . 5 |- ((Fun F /\ Fun G) -> ((dom F = A /\ dom G = B) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B))))
15	14	imp 277	. . . 4 |- (((Fun F /\ Fun G) /\ (dom F = A /\ dom G = B)) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B)))
16	15	an4s 390	. . 3 |- (((Fun F /\ dom F = A) /\ (Fun G /\ dom G = B)) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B)))
17		df-fn 2433	. . 3 |- (F Fn A <-> (Fun F /\ dom F = A))
18		df-fn 2433	. . 3 |- (G Fn B <-> (Fun G /\ dom G = B))
19	16, 17, 18	syl2anb 350	. 2 |- ((F Fn A /\ G Fn B) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B)))
20	19	imp 277	1 |- (((F Fn A /\ G Fn B) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   u. cun 1485   i^i cin 1486  (/)c0 1707  dom cdm 2410  Fun wfun 2416   Fn wfn 2417

This theorem is referenced by:  fun 2763  f1oun 2815

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-fun 2432  df-fn 2433

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