Theorem funun 5147

Description: The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by set.mm contributors, 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		elun 3221	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ (F ∪ G) ↔ (⟨x, y⟩ ∈ F ∨ ⟨x, y⟩ ∈ G))
2		elun 3221	. . . . . . . 8 ⊢ (⟨x, z⟩ ∈ (F ∪ G) ↔ (⟨x, z⟩ ∈ F ∨ ⟨x, z⟩ ∈ G))
3	1, 2	anbi12i 678	. . . . . . 7 ⊢ ((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) ↔ ((⟨x, y⟩ ∈ F ∨ ⟨x, y⟩ ∈ G) ∧ (⟨x, z⟩ ∈ F ∨ ⟨x, z⟩ ∈ G)))
4		anddi 840	. . . . . . 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))))
5	3, 4	bitri 240	. . . . . 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))))
6		sp 1747	. . . . . . . . . 10 ⊢ (∀x(x ∈ dom F → ¬ x ∈ dom G) → (x ∈ dom F → ¬ x ∈ dom G))
7		disj1 3594	. . . . . . . . . 10 ⊢ ((dom F ∩ dom G) = ∅ ↔ ∀x(x ∈ dom F → ¬ x ∈ dom G))
8		imnan 411	. . . . . . . . . . 11 ⊢ ((x ∈ dom F → ¬ x ∈ dom G) ↔ ¬ (x ∈ dom F ∧ x ∈ dom G))
9	8	bicomi 193	. . . . . . . . . 10 ⊢ (¬ (x ∈ dom F ∧ x ∈ dom G) ↔ (x ∈ dom F → ¬ x ∈ dom G))
10	6, 7, 9	3imtr4i 257	. . . . . . . . 9 ⊢ ((dom F ∩ dom G) = ∅ → ¬ (x ∈ dom F ∧ x ∈ dom G))
11		opeldm 4911	. . . . . . . . . 10 ⊢ (⟨x, y⟩ ∈ F → x ∈ dom F)
12		opeldm 4911	. . . . . . . . . 10 ⊢ (⟨x, z⟩ ∈ G → x ∈ dom G)
13	11, 12	anim12i 549	. . . . . . . . 9 ⊢ ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (x ∈ dom F ∧ x ∈ dom G))
14	10, 13	nsyl 113	. . . . . . . 8 ⊢ ((dom F ∩ dom G) = ∅ → ¬ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G))
15		orel2 372	. . . . . . . 8 ⊢ (¬ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G) → (((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) → (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F)))
16	14, 15	syl 15	. . . . . . 7 ⊢ ((dom F ∩ dom G) = ∅ → (((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ G)) → (⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F)))
17		sp 1747	. . . . . . . . . 10 ⊢ (∀x(x ∈ dom G → ¬ x ∈ dom F) → (x ∈ dom G → ¬ x ∈ dom F))
18		incom 3449	. . . . . . . . . . . 12 ⊢ (dom F ∩ dom G) = (dom G ∩ dom F)
19	18	eqeq1i 2360	. . . . . . . . . . 11 ⊢ ((dom F ∩ dom G) = ∅ ↔ (dom G ∩ dom F) = ∅)
20		disj1 3594	. . . . . . . . . . 11 ⊢ ((dom G ∩ dom F) = ∅ ↔ ∀x(x ∈ dom G → ¬ x ∈ dom F))
21	19, 20	bitri 240	. . . . . . . . . 10 ⊢ ((dom F ∩ dom G) = ∅ ↔ ∀x(x ∈ dom G → ¬ x ∈ dom F))
22		imnan 411	. . . . . . . . . . 11 ⊢ ((x ∈ dom G → ¬ x ∈ dom F) ↔ ¬ (x ∈ dom G ∧ x ∈ dom F))
23	22	bicomi 193	. . . . . . . . . 10 ⊢ (¬ (x ∈ dom G ∧ x ∈ dom F) ↔ (x ∈ dom G → ¬ x ∈ dom F))
24	17, 21, 23	3imtr4i 257	. . . . . . . . 9 ⊢ ((dom F ∩ dom G) = ∅ → ¬ (x ∈ dom G ∧ x ∈ dom F))
25		opeldm 4911	. . . . . . . . . 10 ⊢ (⟨x, y⟩ ∈ G → x ∈ dom G)
26		opeldm 4911	. . . . . . . . . 10 ⊢ (⟨x, z⟩ ∈ F → x ∈ dom F)
27	25, 26	anim12i 549	. . . . . . . . 9 ⊢ ((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ F) → (x ∈ dom G ∧ x ∈ dom F))
28	24, 27	nsyl 113	. . . . . . . 8 ⊢ ((dom F ∩ dom G) = ∅ → ¬ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ F))
29		orel1 371	. . . . . . . 8 ⊢ (¬ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ F) → (((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G)) → (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G)))
30	28, 29	syl 15	. . . . . . 7 ⊢ ((dom F ∩ dom G) = ∅ → (((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G)) → (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G)))
31	16, 30	orim12d 811	. . . . . 6 ⊢ ((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))))
32	5, 31	syl5bi 208	. . . . 5 ⊢ ((dom F ∩ dom G) = ∅ → ((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G))))
33		dffun4 5122	. . . . . . . . 9 ⊢ (Fun F ↔ ∀x∀y∀z((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) → y = z))
34	33	biimpi 186	. . . . . . . 8 ⊢ (Fun F → ∀x∀y∀z((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) → y = z))
35	34	19.21bi 1758	. . . . . . 7 ⊢ (Fun F → ∀y∀z((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) → y = z))
36	35	19.21bbi 1865	. . . . . 6 ⊢ (Fun F → ((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) → y = z))
37		dffun4 5122	. . . . . . . . 9 ⊢ (Fun G ↔ ∀x∀y∀z((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G) → y = z))
38	37	biimpi 186	. . . . . . . 8 ⊢ (Fun G → ∀x∀y∀z((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G) → y = z))
39	38	19.21bi 1758	. . . . . . 7 ⊢ (Fun G → ∀y∀z((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G) → y = z))
40	39	19.21bbi 1865	. . . . . 6 ⊢ (Fun G → ((⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G) → y = z))
41	36, 40	jaao 495	. . . . 5 ⊢ ((Fun F ∧ Fun G) → (((⟨x, y⟩ ∈ F ∧ ⟨x, z⟩ ∈ F) ∨ (⟨x, y⟩ ∈ G ∧ ⟨x, z⟩ ∈ G)) → y = z))
42	32, 41	sylan9r 639	. . . 4 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) → y = z))
43	42	alrimiv 1631	. . 3 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ∀z((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) → y = z))
44	43	alrimivv 1632	. 2 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → ∀x∀y∀z((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) → y = z))
45		dffun4 5122	. 2 ⊢ (Fun (F ∪ G) ↔ ∀x∀y∀z((⟨x, y⟩ ∈ (F ∪ G) ∧ ⟨x, z⟩ ∈ (F ∪ G)) → y = z))
46	44, 45	sylibr 203	1 ⊢ (((Fun F ∧ Fun G) ∧ (dom F ∩ dom G) = ∅) → Fun (F ∪ G))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  ⟨cop 4562  dom cdm 4773  Fun wfun 4776

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790

This theorem is used by:  funprg  5150  funprgOLD  5151  fnun  5190  fvun  5379