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| Mirrors > Home > MPE Home > Th. List > fun2d | Structured version Visualization version GIF version | ||
| Description: The union of functions with disjoint domains is a function, deduction version of fun2 6771. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
| Ref | Expression |
|---|---|
| fun2d.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) |
| fun2d.g | ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) |
| fun2d.i | ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) |
| Ref | Expression |
|---|---|
| fun2d | ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fun2d.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) | |
| 2 | fun2d.g | . 2 ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) | |
| 3 | fun2d.i | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) | |
| 4 | fun2 6771 | . 2 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) | |
| 5 | 1, 2, 3, 4 | syl21anc 838 | 1 ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 ⟶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: fresaun 6779 mapunen 9186 ac6sfi 9320 axdc3lem4 10493 fseq1p1m1 13638 uhgrun 29091 upgrun 29135 umgrun 29137 elrspunidl 33456 lbsdiflsp0 33677 reprsuc 34630 dvun 42389 evlselvlem 42596 evlselv 42597 |
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