Theorem fun 4580

Description: The union of two functions with disjoint domains.

Assertion

Ref	Expression
fun	|- (((F:A-->C /\ G:B-->D) /\ (A i^i B) = (/)) -> (F u. G):(A u. B)-->(C u. D))

Proof of Theorem fun

Step	Hyp	Ref	Expression
1		fnun 4520	. . . . 5 |- (((F Fn A /\ G Fn B) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B))
2	1	expcom 403	. . . 4 |- ((A i^i B) = (/) -> ((F Fn A /\ G Fn B) -> (F u. G) Fn (A u. B)))
3		unss12 2778	. . . . . 6 |- ((ran F C_ C /\ ran G C_ D) -> (ran F u. ran G) C_ (C u. D))
4		rnun 4325	. . . . . 6 |- ran ( F u. G) = (ran F u. ran G)
5	3, 4	syl5ss 2661	. . . . 5 |- ((ran F C_ C /\ ran G C_ D) -> ran ( F u. G) C_ (C u. D))
6	5	a1i 8	. . . 4 |- ((A i^i B) = (/) -> ((ran F C_ C /\ ran G C_ D) -> ran ( F u. G) C_ (C u. D)))
7	2, 6	anim12d 617	. . 3 |- ((A i^i B) = (/) -> (((F Fn A /\ G Fn B) /\ (ran F C_ C /\ ran G C_ D)) -> ((F u. G) Fn (A u. B) /\ ran ( F u. G) C_ (C u. D))))
8		df-f 4010	. . . . 5 |- (F:A-->C <-> (F Fn A /\ ran F C_ C))
9		df-f 4010	. . . . 5 |- (G:B-->D <-> (G Fn B /\ ran G C_ D))
10	8, 9	anbi12i 540	. . . 4 |- ((F:A-->C /\ G:B-->D) <-> ((F Fn A /\ ran F C_ C) /\ (G Fn B /\ ran G C_ D)))
11		an4 564	. . . 4 |- (((F Fn A /\ ran F C_ C) /\ (G Fn B /\ ran G C_ D)) <-> ((F Fn A /\ G Fn B) /\ (ran F C_ C /\ ran G C_ D)))
12	10, 11	bitri 190	. . 3 |- ((F:A-->C /\ G:B-->D) <-> ((F Fn A /\ G Fn B) /\ (ran F C_ C /\ ran G C_ D)))
13		df-f 4010	. . 3 |- ((F u. G):(A u. B)-->(C u. D) <-> ((F u. G) Fn (A u. B) /\ ran ( F u. G) C_ (C u. D)))
14	7, 12, 13	3imtr4g 612	. 2 |- ((A i^i B) = (/) -> ((F:A-->C /\ G:B-->D) -> (F u. G):(A u. B)-->(C u. D)))
15	14	impcom 378	1 |- (((F:A-->C /\ G:B-->D) /\ (A i^i B) = (/)) -> (F u. G):(A u. B)-->(C u. D))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  ran crn 3987   Fn wfn 3993  -->wf 3994

This theorem is referenced by:  ac6sfilem3 5508  mapdom2 5588  mapunen 5596

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010

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