Theorem funun 2700

Description: The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43.

Assertion

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

Proof of Theorem funun

Step	Hyp	Ref	Expression
1		funrel 2681	. . . . . 6 ⊢ (Fun F → Rel F)
2		funrel 2681	. . . . . 6 ⊢ (Fun G → Rel G)
3	1, 2	anim12i 268	. . . . 5 ⊢ ((Fun F ∧ Fun G) → (Rel F ∧ Rel G))
4		relun 2490	. . . . 5 ⊢ (Rel (F ∪ G) ↔ (Rel F ∧ Rel G))
5	3, 4	sylibr 175	. . . 4 ⊢ ((Fun F ∧ Fun G) → Rel (F ∪ G))
6	5	adantr 306	. . 3 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → Rel (F ∪ G))
7		disj1 1734	. . . . . . . . . . . . . 14 ⊢ ((dom F ∩ dom G) = ∅ ↔ ∀x(x ∈ dom F → ¬ x ∈ dom G))
8	7	biimp 133	. . . . . . . . . . . . 13 ⊢ ((dom F ∩ dom G) = ∅ → ∀x(x ∈ dom F → ¬ x ∈ dom G))
9	8	19.21bi 742	. . . . . . . . . . . 12 ⊢ ((dom F ∩ dom G) = ∅ → (x ∈ dom F → ¬ x ∈ dom G))
10		imnan 207	. . . . . . . . . . . 12 ⊢ ((x ∈ dom F → ¬ x ∈ dom G) ↔ ¬ (x ∈ dom F ∧ x ∈ dom G))
11	9, 10	sylib 173	. . . . . . . . . . 11 ⊢ ((dom F ∩ dom G) = ∅ → ¬ (x ∈ dom F ∧ x ∈ dom G))
12		visset 1350	. . . . . . . . . . . . 13 ⊢ x ∈ V
13	12	opeldm 2534	. . . . . . . . . . . 12 ⊢ (⟨x, y⟩ ∈ F → x ∈ dom F)
14	12	opeldm 2534	. . . . . . . . . . . 12 ⊢ (⟨x, z⟩ ∈ G → x ∈ dom G)
15	13, 14	anim12i 268	. . . . . . . . . . 11 ⊢ ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (x ∈ dom F ∧ x ∈ dom G))
16	11, 15	nsyl 102	. . . . . . . . . 10 ⊢ ((dom F ∩ dom G) = ∅ → ¬ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G))
17		orel2 213	. . . . . . . . . 10 ⊢ (¬ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) → (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F)))
18	16, 17	syl 12	. . . . . . . . 9 ⊢ ((dom F ∩ dom G) = ∅ → (((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) → (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F)))
19	9	con2d 83	. . . . . . . . . . . 12 ⊢ ((dom F ∩ dom G) = ∅ → (x ∈ dom G → ¬ x ∈ dom F))
20		imnan 207	. . . . . . . . . . . 12 ⊢ ((x ∈ dom G → ¬ x ∈ dom F) ↔ ¬ (x ∈ dom G ∧ x ∈ dom F))
21	19, 20	sylib 173	. . . . . . . . . . 11 ⊢ ((dom F ∩ dom G) = ∅ → ¬ (x ∈ dom G ∧ x ∈ dom F))
22	12	opeldm 2534	. . . . . . . . . . . 12 ⊢ (⟨x, y⟩ ∈ G → x ∈ dom G)
23	12	opeldm 2534	. . . . . . . . . . . 12 ⊢ (⟨x, z⟩ ∈ F → x ∈ dom F)
24	22, 23	anim12i 268	. . . . . . . . . . 11 ⊢ ((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ F) → (x ∈ dom G ∧ x ∈ dom F))
25	21, 24	nsyl 102	. . . . . . . . . 10 ⊢ ((dom F ∩ dom G) = ∅ → ¬ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ F))
26		orel1 212	. . . . . . . . . 10 ⊢ (¬ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ F) → (((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G)) → (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G)))
27	25, 26	syl 12	. . . . . . . . 9 ⊢ ((dom F ∩ dom G) = ∅ → (((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G)) → (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G)))
28	18, 27	orim12d 436	. . . . . . . 8 ⊢ ((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))))
29	28	adantl 305	. . . . . . 7 ⊢ (((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))))
30		elun 1601	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ (F ∪ G) ↔ (⟨x, y⟩ ∈ F ∨ ⟨x, y⟩ ∈ G))
31		elun 1601	. . . . . . . . 9 ⊢ (⟨x, z⟩ ∈ (F ∪ G) ↔ (⟨x, z⟩ ∈ F ∨ ⟨x, z⟩ ∈ G))
32	30, 31	anbi12i 369	. . . . . . . 8 ⊢ ((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) ↔ ((⟨x, y⟩ ∈ F ∨ ⟨x, y⟩ ∈ G) ∧ (⟨x, z⟩ ∈ F ∨ ⟨x, z⟩ ∈ G)))
33		anddi 459	. . . . . . . 8 ⊢ (((⟨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))))
34	32, 33	bitr 151	. . . . . . 7 ⊢ ((⟨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))))
35	29, 34	syl5ib 181	. . . . . 6 ⊢ (((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))))
36		dffun4 2676	. . . . . . . . . . 11 ⊢ (Fun F ↔ (Rel F ∧ ∀x∀y∀z((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) → y = z)))
37	36	pm3.27bd 263	. . . . . . . . . 10 ⊢ (Fun F → ∀x∀y∀z((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) → y = z))
38	37	19.21bi 742	. . . . . . . . 9 ⊢ (Fun F → ∀y∀z((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) → y = z))
39	38	19.21bbi 743	. . . . . . . 8 ⊢ (Fun F → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) → y = z))
40		dffun4 2676	. . . . . . . . . . 11 ⊢ (Fun G ↔ (Rel G ∧ ∀x∀y∀z((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G) → y = z)))
41	40	pm3.27bd 263	. . . . . . . . . 10 ⊢ (Fun G → ∀x∀y∀z((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G) → y = z))
42	41	19.21bi 742	. . . . . . . . 9 ⊢ (Fun G → ∀y∀z((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G) → y = z))
43	42	19.21bbi 743	. . . . . . . 8 ⊢ (Fun G → ((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G) → y = z))
44	39, 43	jaao 330	. . . . . . 7 ⊢ ((Fun F ∧ Fun G) → (((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G)) → y = z))
45	44	adantr 306	. . . . . 6 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → (((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G)) → y = z))
46	35, 45	syld 27	. . . . 5 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) → y = z))
47	46	19.21aiv 943	. . . 4 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ∀z((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) → y = z))
48	47	19.21aivv 944	. . 3 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ∀x∀y∀z((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) → y = z))
49	6, 48	jca 236	. 2 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → (Rel (F ∪ G) ∧ ∀x∀y∀z((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) → y = z)))
50		dffun4 2676	. 2 ⊢ (Fun (F ∪ G) ↔ (Rel (F ∪ G) ∧ ∀x∀y∀z((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) → y = z)))
51	49, 50	sylibr 175	1 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → Fun (F ∪ G))

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196  ∀wal 672   = weq 797   = wceq 1091   ∈ wcel 1092   ∪ cun 1485   ∩ cin 1486  ∅c0 1707  ⟨cop 1810  dom cdm 2410  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  fnun 2730  tfrlem10 2958  sbthlem7 3355  sbthlem8 3356  fodomb 3615

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-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-fun 2432

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