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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 6534. (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 6534 | . 2 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) | |
5 | 1, 2, 3, 4 | syl21anc 833 | 1 ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∪ cun 3931 ∩ cin 3932 ∅c0 4288 ⟶wf 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-fun 6350 df-fn 6351 df-f 6352 |
This theorem is referenced by: fresaun 6542 mapunen 8674 ac6sfi 8750 axdc3lem4 9863 fseq1p1m1 12969 uhgrun 26786 upgrun 26830 umgrun 26832 lbsdiflsp0 30921 reprsuc 31785 |
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