Theorem funun 6546

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 6517	. . . . 5 ⊢ (Fun 𝐹 → Rel 𝐹)
2		funrel 6517	. . . . 5 ⊢ (Fun 𝐺 → Rel 𝐺)
3	1, 2	anim12i 614	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (Rel 𝐹 ∧ Rel 𝐺))
4		relun 5768	. . . 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 4107	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑦⟩ ∈ 𝐺))
8		elun 4107	. . . . . . . 8 ⊢ (⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺) ↔ (⟨𝑥, 𝑧⟩ ∈ 𝐹 ∨ ⟨𝑥, 𝑧⟩ ∈ 𝐺))
9	7, 8	anbi12i 629	. . . . . . 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 4406	. . . . . . . . . . . . 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 2197	. . . . . . . . . . 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 3446	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
18		vex 3446	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
19	17, 18	opeldm 5864	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 → 𝑥 ∈ dom 𝐹)
20		vex 3446	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
21	17, 20	opeldm 5864	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺)
22	19, 21	anim12i 614	. . . . . . . . . 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 5864	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐺 → 𝑥 ∈ dom 𝐺)
30	17, 20	opeldm 5864	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑧⟩ ∈ 𝐹 → 𝑥 ∈ dom 𝐹)
31	29, 30	anim12i 614	. . . . . . . . . 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 6513	. . . . . . . . . 10 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧)))
39	38	simprbi 497	. . . . . . . . 9 ⊢ (Fun 𝐹 → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
40	39	19.21bi 2197	. . . . . . . 8 ⊢ (Fun 𝐹 → ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
41	40	19.21bbi 2198	. . . . . . 7 ⊢ (Fun 𝐹 → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐹) → 𝑦 = 𝑧))
42		dffun4 6513	. . . . . . . . . 10 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑦 = 𝑧)))
43	42	simprbi 497	. . . . . . . . 9 ⊢ (Fun 𝐺 → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑦 = 𝑧))
44	43	19.21bi 2197	. . . . . . . 8 ⊢ (Fun 𝐺 → ∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) → 𝑦 = 𝑧))
45	44	19.21bbi 2198	. . . . . . 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 1929	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∀𝑧((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) → 𝑦 = 𝑧))
50	49	alrimivv 1930	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ (𝐹 ∪ 𝐺) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐹 ∪ 𝐺)) → 𝑦 = 𝑧))
51		dffun4 6513	. 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 584	1 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ∪ cun 3901   ∩ cin 3902  ∅c0 4287  ⟨cop 4588  dom cdm 5632  Rel wrel 5637  Fun wfun 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-fun 6502

This theorem is referenced by:  funprg  6554  funtpg  6555  funtp  6557  funcnvpr  6562  funcnvtp  6563  funcnvqp  6564  fnun  6614  fvun  6932  tfrlem10  8328  sbthlem7  9033  sbthlem8  9034  fodomr  9068  fodomfir  9240  funsnfsupp  9307  axdc3lem4  10375  strleun  17096  setsfun  17110  setsfun0  17111  cnfldfunALT  21336  cnfldfunALTOLD  21349  noextend  27646  noextendseq  27647  bnj1421  35217  satffunlem1  35620  satffunlem2  35621