Theorem fnun 6630

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 6519	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
2		df-fn 6519	. . 3 ⊢ (𝐺 Fn 𝐵 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐵))
3		ineq12 4165	. . . . . . . . . . 11 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
4	3	eqeq1d 2763	. . . . . . . . . 10 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((dom 𝐹 ∩ dom 𝐺) = ∅ ↔ (𝐴 ∩ 𝐵) = ∅))
5	4	anbi2d 639	. . . . . . . . 9 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) ↔ ((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅)))
6		funun 6562	. . . . . . . . 9 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
7	5, 6	biimtrrdi 256	. . . . . . . 8 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅) → Fun (𝐹 ∪ 𝐺)))
8		dmun 5882	. . . . . . . . 9 ⊢ dom (𝐹 ∪ 𝐺) = (dom 𝐹 ∪ dom 𝐺)
9		uneq12 4114	. . . . . . . . 9 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (dom 𝐹 ∪ dom 𝐺) = (𝐴 ∪ 𝐵))
10	8, 9	eqtrid 2808	. . . . . . . 8 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → dom (𝐹 ∪ 𝐺) = (𝐴 ∪ 𝐵))
11	7, 10	jctird 534	. . . . . . 7 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅) → (Fun (𝐹 ∪ 𝐺) ∧ dom (𝐹 ∪ 𝐺) = (𝐴 ∪ 𝐵))))
12		df-fn 6519	. . . . . . 7 ⊢ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵) ↔ (Fun (𝐹 ∪ 𝐺) ∧ dom (𝐹 ∪ 𝐺) = (𝐴 ∪ 𝐵)))
13	11, 12	imbitrrdi 254	. . . . . 6 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
14	13	expd 419	. . . . 5 ⊢ ((dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵) → ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐴 ∩ 𝐵) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))))
15	14	impcom 411	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵)) → ((𝐴 ∩ 𝐵) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
16	15	an4s 670	. . 3 ⊢ (((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ∧ (Fun 𝐺 ∧ dom 𝐺 = 𝐵)) → ((𝐴 ∩ 𝐵) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
17	1, 2, 16	syl2anb 607	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((𝐴 ∩ 𝐵) = ∅ → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)))
18	17	imp 410	1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∪ cun 3900   ∩ cin 3901  ∅c0 4283  dom cdm 5643  Fun wfun 6510   Fn wfn 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-fun 6518  df-fn 6519

This theorem is referenced by:  fnund  6631  fun  6721  foun  6820  f1oun  6821  frrlem11  8271  undifixp  8910  bnj535  35146  fullfunfnv  36257  finixpnum  38065  poimirlem1  38081  poimirlem2  38082  poimirlem3  38083  poimirlem4  38084  poimirlem6  38086  poimirlem7  38087  poimirlem11  38091  poimirlem12  38092  poimirlem16  38096  poimirlem17  38097  poimirlem19  38099  poimirlem20  38100