Theorem funun 5320

Description: The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.)

Assertion

Ref	Expression
funun	⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))

Proof of Theorem funun

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funrel 5293	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
2		funrel 5293	. . . . 5 ⊢ (Fun 𝐺 → Rel 𝐺)
3	1, 2	anim12i 338	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (Rel 𝐹 ∧ Rel 𝐺))
4		relun 4796	. . . 4 ⊢ (Rel (𝐹 ∪ 𝐺) ↔ (Rel 𝐹 ∧ Rel 𝐺))
5	3, 4	sylibr 134	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Rel (𝐹 ∪ 𝐺))
6	5	adantr 276	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Rel (𝐹 ∪ 𝐺))
7		elun 3315	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
8		elun 3315	. . . . . . . 8 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺) ↔ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
9	7, 8	anbi12i 460	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑦⟩ ∈ 𝐺) ∧ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑧⟩ ∈ 𝐺)))
10		anddi 823	. . . . . . 7 ⊢ (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑦⟩ ∈ 𝐺) ∧ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) ↔ (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) ∨ ((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))))
11	9, 10	bitri 184	. . . . . 6 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) ↔ (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) ∨ ((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))))
12		disj1 3512	. . . . . . . . . . . . 13 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ ∀𝑥(𝑥 ∈ dom 𝐹 → ¬ 𝑥 ∈ dom 𝐺))
13	12	biimpi 120	. . . . . . . . . . . 12 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ∀𝑥(𝑥 ∈ dom 𝐹 → ¬ 𝑥 ∈ dom 𝐺))
14	13	19.21bi 1582	. . . . . . . . . . 11 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑥 ∈ dom 𝐹 → ¬ 𝑥 ∈ dom 𝐺))
15		imnan 692	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom 𝐹 → ¬ 𝑥 ∈ dom 𝐺) ↔ ¬ (𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ dom 𝐺))
16	14, 15	sylib 122	. . . . . . . . . 10 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ (𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ dom 𝐺))
17		vex 2776	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
18		vex 2776	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
19	17, 18	opeldm 4886	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ dom 𝐹)
20		vex 2776	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
21	17, 20	opeldm 4886	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺)
22	19, 21	anim12i 338	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ dom 𝐺))
23	16, 22	nsyl 629	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
24		orel2 728	. . . . . . . . 9 ⊢ (¬ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)))
25	23, 24	syl 14	. . . . . . . 8 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹)))
26	14	con2d 625	. . . . . . . . . . 11 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑥 ∈ dom 𝐺 → ¬ 𝑥 ∈ dom 𝐹))
27		imnan 692	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom 𝐺 → ¬ 𝑥 ∈ dom 𝐹) ↔ ¬ (𝑥 ∈ dom 𝐺 ∧ 𝑥 ∈ dom 𝐹))
28	26, 27	sylib 122	. . . . . . . . . 10 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ (𝑥 ∈ dom 𝐺 ∧ 𝑥 ∈ dom 𝐹))
29	17, 18	opeldm 4886	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺)
30	17, 20	opeldm 4886	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑥 ∈ dom 𝐹)
31	29, 30	anim12i 338	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → (𝑥 ∈ dom 𝐺 ∧ 𝑥 ∈ dom 𝐹))
32	28, 31	nsyl 629	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
33		orel1 727	. . . . . . . . 9 ⊢ (¬ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → (((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)))
34	32, 33	syl 14	. . . . . . . 8 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)))
35	25, 34	orim12d 788	. . . . . . 7 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ((((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) ∨ ((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))))
36	35	adantl 277	. . . . . 6 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) ∨ ((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))))
37	11, 36	biimtrid 152	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))))
38		dffun4 5287	. . . . . . . . . 10 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
39	38	simprbi 275	. . . . . . . . 9 ⊢ (Fun 𝐹 → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
40	39	19.21bi 1582	. . . . . . . 8 ⊢ (Fun 𝐹 → ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
41	40	19.21bbi 1583	. . . . . . 7 ⊢ (Fun 𝐹 → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
42		dffun4 5287	. . . . . . . . . 10 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑦 = 𝑧)))
43	42	simprbi 275	. . . . . . . . 9 ⊢ (Fun 𝐺 → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑦 = 𝑧))
44	43	19.21bi 1582	. . . . . . . 8 ⊢ (Fun 𝐺 → ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑦 = 𝑧))
45	44	19.21bbi 1583	. . . . . . 7 ⊢ (Fun 𝐺 → ((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑦 = 𝑧))
46	41, 45	jaao 721	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → 𝑦 = 𝑧))
47	46	adantr 276	. . . . 5 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → 𝑦 = 𝑧))
48	37, 47	syld 45	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) → 𝑦 = 𝑧))
49	48	alrimiv 1898	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∀𝑧((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) → 𝑦 = 𝑧))
50	49	alrimivv 1899	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) → 𝑦 = 𝑧))
51		dffun4 5287	. 2 ⊢ (Fun (𝐹 ∪ 𝐺) ↔ (Rel (𝐹 ∪ 𝐺) ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) → 𝑦 = 𝑧)))
52	6, 50, 51	sylanbrc 417	1 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710  ∀wal 1371   = wceq 1373   ∈ wcel 2177   ∪ cun 3165   ∩ cin 3166  ∅c0 3461  ⟨cop 3637  dom cdm 4679  Rel wrel 4684  Fun wfun 5270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-id 4344  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-fun 5278

This theorem is referenced by:  funprg  5329  funtpg  5330  funtp  5332  fnun  5387  fvun1  5652  sbthlem7  7072  sbthlemi8  7073  casefun  7194  caseinj  7198  djufun  7213  djuinj  7215  exmidfodomrlemim  7316  setsfun  12911  setsfun0  12912  strleund  12979  strleun  12980