| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > fun | Structured version Visualization version 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 6682 | . . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵)) | |
| 2 | 1 | expcom 413 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))) |
| 3 | rnun 6165 | . . . . 5 ⊢ ran (𝐹 ∪ 𝐺) = (ran 𝐹 ∪ ran 𝐺) | |
| 4 | unss12 4188 | . . . . 5 ⊢ ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → (ran 𝐹 ∪ ran 𝐺) ⊆ (𝐶 ∪ 𝐷)) | |
| 5 | 3, 4 | eqsstrid 4022 | . . . 4 ⊢ ((ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷) → ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷)) |
| 6 | 2, 5 | anim12d1 610 | . . 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)))) |
| 7 | df-f 6565 | . . . . 5 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)) | |
| 8 | df-f 6565 | . . . . 5 ⊢ (𝐺:𝐵⟶𝐷 ↔ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐷)) | |
| 9 | 7, 8 | anbi12i 628 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ↔ ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶) ∧ (𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐷))) |
| 10 | an4 656 | . . . 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))) | |
| 11 | 9, 10 | bitri 275 | . . 3 ⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ↔ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (ran 𝐹 ⊆ 𝐶 ∧ ran 𝐺 ⊆ 𝐷))) |
| 12 | df-f 6565 | . . 3 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷) ↔ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵) ∧ ran (𝐹 ∪ 𝐺) ⊆ (𝐶 ∪ 𝐷))) | |
| 13 | 6, 11, 12 | 3imtr4g 296 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷))) |
| 14 | 13 | impcom 407 | 1 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ran crn 5686 Fn wfn 6556 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: fun2 6771 ftpg 7176 fsnunf 7205 ralxpmap 8936 hashfxnn0 14376 cats1un 14759 pwssplit1 21058 axlowdimlem10 28966 wlkp1 29699 padct 32731 eulerpartlemt 34373 sseqf 34394 poimirlem3 37630 poimirlem16 37643 poimirlem19 37646 poimirlem22 37649 poimirlem23 37650 poimirlem24 37651 poimirlem25 37652 poimirlem28 37655 poimirlem29 37656 poimirlem31 37658 mapfzcons 42727 diophrw 42770 diophren 42824 pwssplit4 43101 aacllem 49320 |
| Copyright terms: Public domain | W3C validator |