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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 6543. (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 6543 | . 2 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) | |
5 | 1, 2, 3, 4 | syl21anc 835 | 1 ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 ⟶wf 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 |
This theorem is referenced by: fresaun 6551 mapunen 8688 ac6sfi 8764 axdc3lem4 9877 fseq1p1m1 12984 uhgrun 26861 upgrun 26905 umgrun 26907 lbsdiflsp0 31024 reprsuc 31888 |
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