![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6515. (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 6515 | . 2 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) | |
5 | 1, 2, 3, 4 | syl21anc 836 | 1 ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 ⟶wf 6320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 |
This theorem is referenced by: fresaun 6523 mapunen 8670 ac6sfi 8746 axdc3lem4 9864 fseq1p1m1 12976 uhgrun 26867 upgrun 26911 umgrun 26913 elrspunidl 31014 lbsdiflsp0 31110 reprsuc 31996 |
Copyright terms: Public domain | W3C validator |