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Mathbox for Thierry Arnoux |
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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 34232 | . . . 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 8888 | . 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 2106 ∀wral 3059 ∪ cuni 4912 ◡ccnv 5688 ran crn 5690 “ cima 5692 ⟶wf 6559 (class class class)co 7431 ↑m cmap 8865 sigAlgebracsiga 34089 MblFnMcmbfm 34230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 df-mbfm 34231 |
This theorem is referenced by: imambfm 34244 mbfmco 34246 mbfmco2 34247 mbfmvolf 34248 sibff 34318 sitgclg 34324 orvcval4 34442 |
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