Theorem fnun 6660

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 6543	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2		df-fn 6543	. . 3 ⊢ (𝐺 Fn 𝐵 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐵))
3		ineq12 4206	. . . . . . . . . . 11 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
4	3	eqeq1d 2734	. . . . . . . . . 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 6591	. . . . . . . . 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 5908	. . . . . . . . 9 ⊢ dom (𝐹 ∪ 𝐺) = (dom 𝐹 ∪ dom 𝐺)
9		uneq12 4157	. . . . . . . . 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 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 6543	. . . . . . 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 658	. . 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 1541   ∪ cun 3945   ∩ cin 3946  ∅c0 4321  dom cdm 5675  Fun wfun 6534   Fn wfn 6535

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-fun 6542  df-fn 6543

This theorem is referenced by:  fnund  6661  fun  6750  foun  6848  f1oun  6849  frrlem11  8277  undifixp  8924  bnj535  33889  fullfunfnv  34906  finixpnum  36461  poimirlem1  36477  poimirlem2  36478  poimirlem3  36479  poimirlem4  36480  poimirlem6  36482  poimirlem7  36483  poimirlem11  36487  poimirlem12  36488  poimirlem16  36492  poimirlem17  36493  poimirlem19  36495  poimirlem20  36496