![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfdmssre | Structured version Visualization version GIF version |
Description: The domain of a measurable function is a subset of the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
mbfdmssre | ⊢ (𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismbf1 25654 | . . 3 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))) | |
2 | 1 | simplbi 497 | . 2 ⊢ (𝐹 ∈ MblFn → 𝐹 ∈ (ℂ ↑pm ℝ)) |
3 | elpmi2 45118 | . 2 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → dom 𝐹 ⊆ ℝ) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2104 ∀wral 3057 ⊆ wss 3963 ◡ccnv 5682 dom cdm 5683 ran crn 5684 “ cima 5686 ∘ ccom 5687 (class class class)co 7425 ↑pm cpm 8860 ℂcc 11144 ℝcr 11145 (,)cioo 13377 ℜcre 15122 ℑcim 15123 volcvol 25493 MblFncmbf 25644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-fv 6566 df-ov 7428 df-oprab 7429 df-mpo 7430 df-1st 8007 df-2nd 8008 df-pm 8862 df-mbf 25649 |
This theorem is referenced by: mbfresmf 46645 |
Copyright terms: Public domain | W3C validator |