Theorem fun 5006

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

Assertion

Ref	Expression
fun	⊢ (((𝐹:A⟶𝐶 ∧ 𝐺:B⟶𝐷) ∧ (A ∩ B) = ∅) → (𝐹 ∪ 𝐺):(A ∪ B)⟶(𝐶 ∪ 𝐷))

Proof of Theorem fun

Step	Hyp	Ref	Expression
1		fnun 4948	. . . . 5 ⊢ (((𝐹 Fn A ∧ 𝐺 Fn B) ∧ (A ∩ B) = ∅) → (𝐹 ∪ 𝐺) Fn (A ∪ B))
2	1	expcom 109	. . . 4 ⊢ ((A ∩ B) = ∅ → ((𝐹 Fn A ∧ 𝐺 Fn B) → (𝐹 ∪ 𝐺) Fn (A ∪ B)))
3		rnun 4675	. . . . . 6 ⊢ ran (𝐹 ∪ 𝐺) = (ran 𝐹 ∪ ran 𝐺)
4		unss12 3109	. . . . . 6 ⊢ ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶 ∪ 𝐷))
5	3, 4	syl5eqss 2983	. . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷))
6	5	a1i 9	. . . 4 ⊢ ((A ∩ B) = ∅ → ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷)))
7	2, 6	anim12d 318	. . 3 ⊢ ((A ∩ B) = ∅ → (((𝐹 Fn A ∧ 𝐺 Fn B) ∧ (ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷)) → ((𝐹 ∪ 𝐺) Fn (A ∪ B) ∧ ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷))))
8		df-f 4849	. . . . 5 ⊢ (𝐹:A⟶𝐶 ↔ (𝐹 Fn A ∧ ran 𝐹 ⊆ 𝐶))
9		df-f 4849	. . . . 5 ⊢ (𝐺:B⟶𝐷 ↔ (𝐺 Fn B ∧ ran 𝐺 ⊆ 𝐷))
10	8, 9	anbi12i 433	. . . 4 ⊢ ((𝐹:A⟶𝐶 ∧ 𝐺:B⟶𝐷) ↔ ((𝐹 Fn A ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn B ∧ ran 𝐺 ⊆ 𝐷)))
11		an4 520	. . . 4 ⊢ (((𝐹 Fn A ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn B ∧ ran 𝐺 ⊆ 𝐷)) ↔ ((𝐹 Fn A ∧ 𝐺 Fn B) ∧ (ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷)))
12	10, 11	bitri 173	. . 3 ⊢ ((𝐹:A⟶𝐶 ∧ 𝐺:B⟶𝐷) ↔ ((𝐹 Fn A ∧ 𝐺 Fn B) ∧ (ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷)))
13		df-f 4849	. . 3 ⊢ ((𝐹 ∪ 𝐺):(A ∪ B)⟶(𝐶 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (A ∪ B) ∧ ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷)))
14	7, 12, 13	3imtr4g 194	. 2 ⊢ ((A ∩ B) = ∅ → ((𝐹:A⟶𝐶 ∧ 𝐺:B⟶𝐷) → (𝐹 ∪ 𝐺):(A ∪ B)⟶(𝐶 ∪ 𝐷)))
15	14	impcom 116	1 ⊢ (((𝐹:A⟶𝐶 ∧ 𝐺:B⟶𝐷) ∧ (A ∩ B) = ∅) → (𝐹 ∪ 𝐺):(A ∪ B)⟶(𝐶 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∪ cun 2909   ∩ cin 2910   ⊆ wss 2911  ∅c0 3218  ran crn 4289   Fn wfn 4840  ⟶wf 4841

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849

This theorem is referenced by:  fun2  5007  ftpg  5290  fsnunf  5305