![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > fun | GIF version |
Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.) |
Ref | Expression |
---|---|
fun | ⊢ (((𝐹:A⟶𝐶 ∧ 𝐺:B⟶𝐷) ∧ (A ∩ B) = ∅) → (𝐹 ∪ 𝐺):(A ∪ B)⟶(𝐶 ∪ 𝐷)) |
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)) |
Colors of variables: wff set class |
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-bndl 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 |
Copyright terms: Public domain | W3C validator |