Theorem fnun 6606

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 6495	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2		df-fn 6495	. . 3 ⊢ (𝐺 Fn 𝐵 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐵))
3		ineq12 4167	. . . . . . . . . . 11 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
4	3	eqeq1d 2738	. . . . . . . . . 10 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
5	4	anbi2d 630	. . . . . . . . 9 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ↔ ((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅)))
6		funun 6538	. . . . . . . . 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 5859	. . . . . . . . 9 ⊢ dom (𝐹 ∪ 𝐺) = (dom 𝐹 ∪ dom 𝐺)
9		uneq12 4115	. . . . . . . . 9 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∪ dom 𝐺) = (𝐴 ∪ 𝐵))
10	8, 9	eqtrid 2783	. . . . . . . 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 6495	. . . . . . 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 660	. . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ (Fun 𝐺 ∧ dom 𝐺 = 𝐵)) → ((𝐴 ∩ 𝐵) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
17	1, 2, 16	syl2anb 598	. 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 1541   ∪ cun 3899   ∩ cin 3900  ∅c0 4285  dom cdm 5624  Fun wfun 6486   Fn wfn 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-fun 6494  df-fn 6495

This theorem is referenced by:  fnund  6607  fun  6696  foun  6792  f1oun  6793  frrlem11  8238  undifixp  8872  bnj535  35046  fullfunfnv  36140  finixpnum  37802  poimirlem1  37818  poimirlem2  37819  poimirlem3  37820  poimirlem4  37821  poimirlem6  37823  poimirlem7  37824  poimirlem11  37828  poimirlem12  37829  poimirlem16  37833  poimirlem17  37834  poimirlem19  37836  poimirlem20  37837