Theorem fnun 3594

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 2212	. . . . . . . . . . 11 |- ((dom F = A /\ dom G = B) -> (dom F i^i dom G) = (A i^i B))
2	1	eqeq1d 1483	. . . . . . . . . 10 |- ((dom F = A /\ dom G = B) -> ((dom F i^i dom G) = (/) <-> (A i^i B) = (/)))
3	2	anbi2d 616	. . . . . . . . 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 3554	. . . . . . . . 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 2179	. . . . . . . . 9 |- ((dom F = A /\ dom G = B) -> (dom F u. dom G) = (A u. B))
7		dmun 3317	. . . . . . . . 9 |- dom ( F u. G) = (dom F u. dom G)
8	6, 7	syl5eq 1519	. . . . . . . 8 |- ((dom F = A /\ dom G = B) -> dom ( F u. G) = (A u. B))
9	5, 8	jctird 602	. . . . . . 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 3193	. . . . . . 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 3193	. . 3 |- (F Fn A <-> (Fun F /\ dom F = A))
16		df-fn 3193	. . 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 956   u. cun 2045   i^i cin 2046  (/)c0 2280  dom cdm 3170  Fun wfun 3176   Fn wfn 3177

This theorem is referenced by:  fun 3641  f1oun 3706

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-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  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-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-fun 3192  df-fn 3193

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