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Mirrors > Home > MPE Home > Th. List > fun2 | Structured version Visualization version GIF version |
Description: The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.) |
Ref | Expression |
---|---|
fun2 | ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fun 6227 | . 2 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐶)) | |
2 | unidm 3899 | . . 3 ⊢ (𝐶 ∪ 𝐶) = 𝐶 | |
3 | feq3 6189 | . . 3 ⊢ ((𝐶 ∪ 𝐶) = 𝐶 → ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐶) ↔ (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐶) ↔ (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) |
5 | 1, 4 | sylib 208 | 1 ⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∪ cun 3713 ∩ cin 3714 ∅c0 4058 ⟶wf 6045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-id 5174 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-fun 6051 df-fn 6052 df-f 6053 |
This theorem is referenced by: fun2d 6229 fresaun 6236 mapunen 8296 ac6sfi 8371 axdc3lem4 9487 fseq1p1m1 12627 axlowdimlem5 26046 axlowdimlem7 26048 resf1o 29835 locfinref 30238 breprexplema 31038 |
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