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| Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfmf | Structured version Visualization version GIF version | ||
| Description: A measurable function as a function with domain and codomain. (Contributed by Thierry Arnoux, 25-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| mbfmf.1 | ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) | 
| mbfmf.2 | ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) | 
| mbfmf.3 | ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) | 
| Ref | Expression | 
|---|---|
| mbfmf | ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mbfmf.3 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) | |
| 2 | mbfmf.1 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) | |
| 3 | mbfmf.2 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) | |
| 4 | 2, 3 | ismbfm 34252 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))) | 
| 5 | 1, 4 | mpbid 232 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) | 
| 6 | 5 | simpld 494 | . 2 ⊢ (𝜑 → 𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆)) | 
| 7 | elmapi 8889 | . 2 ⊢ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) → 𝐹:∪ 𝑆⟶∪ 𝑇) | |
| 8 | 6, 7 | syl 17 | 1 ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ∪ cuni 4907 ◡ccnv 5684 ran crn 5686 “ cima 5688 ⟶wf 6557 (class class class)co 7431 ↑m cmap 8866 sigAlgebracsiga 34109 MblFnMcmbfm 34250 | 
| 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-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| 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-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-mbfm 34251 | 
| This theorem is referenced by: imambfm 34264 mbfmco 34266 mbfmco2 34267 mbfmvolf 34268 sibff 34338 sitgclg 34344 orvcval4 34463 | 
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