Theorem funun 6612

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 6583	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
2		funrel 6583	. . . . 5 ⊢ (Fun 𝐺 → Rel 𝐺)
3	1, 2	anim12i 613	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (Rel 𝐹 ∧ Rel 𝐺))
4		relun 5821	. . . 4 ⊢ (Rel (𝐹 ∪ 𝐺) ↔ (Rel 𝐹 ∧ Rel 𝐺))
5	3, 4	sylibr 234	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Rel (𝐹 ∪ 𝐺))
6	5	adantr 480	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Rel (𝐹 ∪ 𝐺))
7		elun 4153	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
8		elun 4153	. . . . . . . 8 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺) ↔ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
9	7, 8	anbi12i 628	. . . . . . 7 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑦⟩ ∈ 𝐺) ∧ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑧⟩ ∈ 𝐺)))
10		anddi 1013	. . . . . . 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 275	. . . . . 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 4452	. . . . . . . . . . . . 13 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ ∀𝑥(𝑥 ∈ dom 𝐹 → ¬ 𝑥 ∈ dom 𝐺))
13	12	biimpi 216	. . . . . . . . . . . 12 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ∀𝑥(𝑥 ∈ dom 𝐹 → ¬ 𝑥 ∈ dom 𝐺))
14	13	19.21bi 2189	. . . . . . . . . . 11 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑥 ∈ dom 𝐹 → ¬ 𝑥 ∈ dom 𝐺))
15		imnan 399	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom 𝐹 → ¬ 𝑥 ∈ dom 𝐺) ↔ ¬ (𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ dom 𝐺))
16	14, 15	sylib 218	. . . . . . . . . 10 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ (𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ dom 𝐺))
17		vex 3484	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
18		vex 3484	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
19	17, 18	opeldm 5918	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ dom 𝐹)
20		vex 3484	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
21	17, 20	opeldm 5918	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺)
22	19, 21	anim12i 613	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → (𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ dom 𝐺))
23	16, 22	nsyl 140	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
24		orel2 891	. . . . . . . . 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 17	. . . . . . . 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 134	. . . . . . . . . . 11 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑥 ∈ dom 𝐺 → ¬ 𝑥 ∈ dom 𝐹))
27		imnan 399	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ dom 𝐺 → ¬ 𝑥 ∈ dom 𝐹) ↔ ¬ (𝑥 ∈ dom 𝐺 ∧ 𝑥 ∈ dom 𝐹))
28	26, 27	sylib 218	. . . . . . . . . 10 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ (𝑥 ∈ dom 𝐺 ∧ 𝑥 ∈ dom 𝐹))
29	17, 18	opeldm 5918	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺)
30	17, 20	opeldm 5918	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑥 ∈ dom 𝐹)
31	29, 30	anim12i 613	. . . . . . . . . 10 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → (𝑥 ∈ dom 𝐺 ∧ 𝑥 ∈ dom 𝐹))
32	28, 31	nsyl 140	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹))
33		orel1 889	. . . . . . . . 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 17	. . . . . . . 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 967	. . . . . . 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 481	. . . . . 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 242	. . . . 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 6577	. . . . . . . . . 10 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
39	38	simprbi 496	. . . . . . . . 9 ⊢ (Fun 𝐹 → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
40	39	19.21bi 2189	. . . . . . . 8 ⊢ (Fun 𝐹 → ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
41	40	19.21bbi 2190	. . . . . . 7 ⊢ (Fun 𝐹 → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
42		dffun4 6577	. . . . . . . . . 10 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑦 = 𝑧)))
43	42	simprbi 496	. . . . . . . . 9 ⊢ (Fun 𝐺 → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑦 = 𝑧))
44	43	19.21bi 2189	. . . . . . . 8 ⊢ (Fun 𝐺 → ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑦 = 𝑧))
45	44	19.21bbi 2190	. . . . . . 7 ⊢ (Fun 𝐺 → ((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑦 = 𝑧))
46	41, 45	jaao 957	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) ∨ (⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺)) → 𝑦 = 𝑧))
47	46	adantr 480	. . . . 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 47	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) → 𝑦 = 𝑧))
49	48	alrimiv 1927	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∀𝑧((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) → 𝑦 = 𝑧))
50	49	alrimivv 1928	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) → 𝑦 = 𝑧))
51		dffun4 6577	. 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 583	1 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848  ∀wal 1538   = wceq 1540   ∈ wcel 2108   ∪ cun 3949   ∩ cin 3950  ∅c0 4333  ⟨cop 4632  dom cdm 5685  Rel wrel 5690  Fun wfun 6555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-fun 6563

This theorem is referenced by:  funprg  6620  funtpg  6621  funtp  6623  funcnvpr  6628  funcnvtp  6629  funcnvqp  6630  fnun  6682  fvun  6999  wfrlem13OLD  8361  tfrlem10  8427  sbthlem7  9129  sbthlem8  9130  fodomr  9168  fodomfir  9368  funsnfsupp  9432  axdc3lem4  10493  strleun  17194  setsfun  17208  setsfun0  17209  cnfldfunALT  21379  cnfldfunALTOLD  21392  cnfldfunALTOLDOLD  21393  noextend  27711  noextendseq  27712  bnj1421  35056  satffunlem1  35412  satffunlem2  35413