Theorem fun 3641

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 3594	. . . . 5 |- (((F Fn A /\ G Fn B) /\ (A i^i B) = (/)) -> (F u. G) Fn (A u. B))
2	1	expcom 374	. . . 4 |- ((A i^i B) = (/) -> ((F Fn A /\ G Fn B) -> (F u. G) Fn (A u. B)))
3		unss12 2202	. . . . . 6 |- ((ran F (_ C /\ ran G (_ D) -> (ran F u. ran G) (_ (C u. D))
4		rnun 3457	. . . . . 6 |- ran ( F u. G) = (ran F u. ran G)
5	3, 4	syl5ss 2105	. . . . 5 |- ((ran F (_ C /\ ran G (_ D) -> ran ( F u. G) (_ (C u. D))
6	5	a1i 8	. . . 4 |- ((A i^i B) = (/) -> ((ran F (_ C /\ ran G (_ D) -> ran ( F u. G) (_ (C u. D)))
7	2, 6	anim12d 558	. . 3 |- ((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))))
8		df-f 3194	. . . . 5 |- (F:A-->C <-> (F Fn A /\ ran F (_ C))
9		df-f 3194	. . . . 5 |- (G:B-->D <-> (G Fn B /\ ran G (_ D))
10	8, 9	anbi12i 482	. . . 4 |- ((F:A-->C /\ G:B-->D) <-> ((F Fn A /\ ran F (_ C) /\ (G Fn B /\ ran G (_ D)))
11		an4 506	. . . 4 |- (((F Fn A /\ ran F (_ C) /\ (G Fn B /\ ran G (_ D)) <-> ((F Fn A /\ G Fn B) /\ (ran F (_ C /\ ran G (_ D)))
12	10, 11	bitr 173	. . 3 |- ((F:A-->C /\ G:B-->D) <-> ((F Fn A /\ G Fn B) /\ (ran F (_ C /\ ran G (_ D)))
13		df-f 3194	. . 3 |- ((F u. G):(A u. B)-->(C u. D) <-> ((F u. G) Fn (A u. B) /\ ran ( F u. G) (_ (C u. D)))
14	7, 12, 13	3imtr4g 553	. 2 |- ((A i^i B) = (/) -> ((F:A-->C /\ G:B-->D) -> (F u. G):(A u. B)-->(C u. D)))
15	14	impcom 351	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 223   = wceq 956   u. cun 2045   i^i cin 2046   (_ wss 2047  (/)c0 2280  ran crn 3171   Fn wfn 3177  -->wf 3178

This theorem is referenced by:  mapdom2 4494  mapunen 4502

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-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194

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