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 24217 | . . 3 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))) | |
2 | 1 | simplbi 500 | . 2 ⊢ (𝐹 ∈ MblFn → 𝐹 ∈ (ℂ ↑pm ℝ)) |
3 | elpmi2 41479 | . 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 398 ∈ wcel 2108 ∀wral 3136 ⊆ wss 3934 ◡ccnv 5547 dom cdm 5548 ran crn 5549 “ cima 5551 ∘ ccom 5552 (class class class)co 7148 ↑pm cpm 8399 ℂcc 10527 ℝcr 10528 (,)cioo 12730 ℜcre 14448 ℑcim 14449 volcvol 24056 MblFncmbf 24207 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-pm 8401 df-mbf 24212 |
This theorem is referenced by: mbfresmf 43007 |
Copyright terms: Public domain | W3C validator |