Theorem fnun 6545

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 6436	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2		df-fn 6436	. . 3 ⊢ (𝐺 Fn 𝐵 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐵))
3		ineq12 4141	. . . . . . . . . . 11 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
4	3	eqeq1d 2740	. . . . . . . . . 10 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
5	4	anbi2d 629	. . . . . . . . 9 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ↔ ((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅)))
6		funun 6480	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
7	5, 6	syl6bir 253	. . . . . . . 8 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅) → Fun (𝐹 ∪ 𝐺)))
8		dmun 5819	. . . . . . . . 9 ⊢ dom (𝐹 ∪ 𝐺) = (dom 𝐹 ∪ dom 𝐺)
9		uneq12 4092	. . . . . . . . 9 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∪ dom 𝐺) = (𝐴 ∪ 𝐵))
10	8, 9	eqtrid 2790	. . . . . . . 8 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → dom (𝐹 ∪ 𝐺) = (𝐴 ∪ 𝐵))
11	7, 10	jctird 527	. . . . . . 7 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅) → (Fun (𝐹 ∪ 𝐺) ∧ dom (𝐹 ∪ 𝐺) = (𝐴 ∪ 𝐵))))
12		df-fn 6436	. . . . . . 7 ⊢ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵) ↔ (Fun (𝐹 ∪ 𝐺) ∧ dom (𝐹 ∪ 𝐺) = (𝐴 ∪ 𝐵)))
13	11, 12	syl6ibr 251	. . . . . 6 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
14	13	expd 416	. . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐴 ∩ 𝐵) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))))
15	14	impcom 408	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵)) → ((𝐴 ∩ 𝐵) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
16	15	an4s 657	. . 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 407	1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∪ cun 3885   ∩ cin 3886  ∅c0 4256  dom cdm 5589  Fun wfun 6427   Fn wfn 6428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-fun 6435  df-fn 6436

This theorem is referenced by:  fnund  6546  fun  6636  foun  6734  f1oun  6735  frrlem11  8112  undifixp  8722  bnj535  32870  fullfunfnv  34248  finixpnum  35762  poimirlem1  35778  poimirlem2  35779  poimirlem3  35780  poimirlem4  35781  poimirlem6  35783  poimirlem7  35784  poimirlem11  35788  poimirlem12  35789  poimirlem16  35793  poimirlem17  35794  poimirlem19  35796  poimirlem20  35797