Theorem fun 2763

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 2730	. . . . . . 7 |- (((F Fn A /\ G Fn B) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B))
2	1	exp 291	. . . . . 6 |- ((F Fn A /\ G Fn B) -> ((A i^i B) = (/) -> (F u. G) Fn (A u. B)))
3	2	com12 13	. . . . 5 |- ((A i^i B) = (/) -> ((F Fn A /\ G Fn B) -> (F u. G) Fn (A u. B)))
4		unss12 1630	. . . . . . 7 |- ((ran F (_ C /\ ran G (_ D) -> (ran F u. ran G) (_ (C u. D))
5		rnun 2644	. . . . . . 7 |- ran (F u. G) = (ran F u. ran G)
6	4, 5	syl5ss 1544	. . . . . 6 |- ((ran F (_ C /\ ran G (_ D) -> ran (F u. G) (_ (C u. D))
7	6	a1i 7	. . . . 5 |- ((A i^i B) = (/) -> ((ran F (_ C /\ ran G (_ D) -> ran (F u. G) (_ (C u. D)))
8	3, 7	anim12d 431	. . . 4 |- ((A i^i B) = (/) -> (((F Fn A /\ G Fn B) /\ (ran F (_ C /\ ran G (_ D)) -> ((F u. G) Fn (A u. B) /\ ran (F u. G) (_ (C u. D))))
9		df-f 2434	. . . . . 6 |- (F:A-->C <-> (F Fn A /\ ran F (_ C))
10		df-f 2434	. . . . . 6 |- (G:B-->D <-> (G Fn B /\ ran G (_ D))
11	9, 10	anbi12i 369	. . . . 5 |- ((F:A-->C /\ G:B-->D) <-> ((F Fn A /\ ran F (_ C) /\ (G Fn B /\ ran G (_ D)))
12		an4 388	. . . . 5 |- (((F Fn A /\ ran F (_ C) /\ (G Fn B /\ ran G (_ D)) <-> ((F Fn A /\ G Fn B) /\ (ran F (_ C /\ ran G (_ D)))
13	11, 12	bitr 151	. . . 4 |- ((F:A-->C /\ G:B-->D) <-> ((F Fn A /\ G Fn B) /\ (ran F (_ C /\ ran G (_ D)))
14		df-f 2434	. . . 4 |- ((F u. G):(A u. B)-->(C u. D) <-> ((F u. G) Fn (A u. B) /\ ran (F u. G) (_ (C u. D)))
15	8, 13, 14	3imtr4g 426	. . 3 |- ((A i^i B) = (/) -> ((F:A-->C /\ G:B-->D) -> (F u. G):(A u. B)-->(C u. D)))
16	15	com12 13	. 2 |- ((F:A-->C /\ G:B-->D) -> ((A i^i B) = (/) -> (F u. G):(A u. B)-->(C u. D)))
17	16	imp 277	1 |- (((F:A-->C /\ G:B-->D) /\ (A i^i B) = (/)) -> (F u. G):(A u. B)-->(C u. D))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   u. cun 1485   i^i cin 1486   (_ wss 1487  (/)c0 1707  ran crn 2411   Fn wfn 2417  -->wf 2418

This theorem is referenced by:  mapdom2 3389  mapunen 3397

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-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434

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