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Theorem mbfdmssre 44331
Description: The domain of a measurable function is a subset of the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Assertion
Ref Expression
mbfdmssre (𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ)

Proof of Theorem mbfdmssre
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ismbf1 25011 . . 3 (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)(((ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ ((ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
21simplbi 499 . 2 (𝐹 ∈ MblFn → 𝐹 ∈ (ℂ ↑pm ℝ))
3 elpmi2 43537 . 2 (𝐹 ∈ (ℂ ↑pm ℝ) → dom 𝐹 ⊆ ℝ)
42, 3syl 17 1 (𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wral 3061  wss 3914  ccnv 5636  dom cdm 5637  ran crn 5638  cima 5640  ccom 5641  (class class class)co 7361  pm cpm 8772  cc 11057  cr 11058  (,)cioo 13273  cre 14991  cim 14992  volcvol 24850  MblFncmbf 25001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-pm 8774  df-mbf 25006
This theorem is referenced by:  mbfresmf  45070
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