Theorem fnun 6616

Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)

Assertion

Ref	Expression
fnun	⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))

Proof of Theorem fnun

Step	Hyp	Ref	Expression
1		df-fn 6505	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2		df-fn 6505	. . 3 ⊢ (𝐺 Fn 𝐵 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐵))
3		ineq12 4169	. . . . . . . . . . 11 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
4	3	eqeq1d 2739	. . . . . . . . . 10 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
5	4	anbi2d 631	. . . . . . . . 9 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ↔ ((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅)))
6		funun 6548	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
7	5, 6	biimtrrdi 254	. . . . . . . 8 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅) → Fun (𝐹 ∪ 𝐺)))
8		dmun 5869	. . . . . . . . 9 ⊢ dom (𝐹 ∪ 𝐺) = (dom 𝐹 ∪ dom 𝐺)
9		uneq12 4117	. . . . . . . . 9 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∪ dom 𝐺) = (𝐴 ∪ 𝐵))
10	8, 9	eqtrid 2784	. . . . . . . 8 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → dom (𝐹 ∪ 𝐺) = (𝐴 ∪ 𝐵))
11	7, 10	jctird 526	. . . . . . 7 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅) → (Fun (𝐹 ∪ 𝐺) ∧ dom (𝐹 ∪ 𝐺) = (𝐴 ∪ 𝐵))))
12		df-fn 6505	. . . . . . 7 ⊢ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵) ↔ (Fun (𝐹 ∪ 𝐺) ∧ dom (𝐹 ∪ 𝐺) = (𝐴 ∪ 𝐵)))
13	11, 12	imbitrrdi 252	. . . . . 6 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
14	13	expd 415	. . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐴 ∩ 𝐵) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))))
15	14	impcom 407	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵)) → ((𝐴 ∩ 𝐵) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
16	15	an4s 661	. . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ (Fun 𝐺 ∧ dom 𝐺 = 𝐵)) → ((𝐴 ∩ 𝐵) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
17	1, 2, 16	syl2anb 599	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐴 ∩ 𝐵) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
18	17	imp 406	1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∪ cun 3901   ∩ cin 3902  ∅c0 4287  dom cdm 5634  Fun wfun 6496   Fn wfn 6497

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 5245  ax-pr 5381

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 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-fun 6504  df-fn 6505

This theorem is referenced by:  fnund  6617  fun  6706  foun  6802  f1oun  6803  frrlem11  8250  undifixp  8886  bnj535  35072  fullfunfnv  36168  finixpnum  37885  poimirlem1  37901  poimirlem2  37902  poimirlem3  37903  poimirlem4  37904  poimirlem6  37906  poimirlem7  37907  poimirlem11  37911  poimirlem12  37912  poimirlem16  37916  poimirlem17  37917  poimirlem19  37919  poimirlem20  37920