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Theorem frexr 42016
Description: A function taking real values, is a function taking extended real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
frexr.1 (𝜑𝐹:𝐴⟶ℝ)
Assertion
Ref Expression
frexr (𝜑𝐹:𝐴⟶ℝ*)

Proof of Theorem frexr
StepHypRef Expression
1 frexr.1 . 2 (𝜑𝐹:𝐴⟶ℝ)
2 ressxr 10678 . . 3 ℝ ⊆ ℝ*
32a1i 11 . 2 (𝜑 → ℝ ⊆ ℝ*)
41, 3fssd 6506 1 (𝜑𝐹:𝐴⟶ℝ*)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3884  wf 6324  cr 10529  *cxr 10667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-f 6332  df-xr 10672
This theorem is referenced by:  limsupubuz  42352  limsupreuz  42376  limsupvaluz2  42377  supcnvlimsup  42379  limsupgtlem  42416  liminflimsupclim  42446  climliminflimsup2  42448  climliminflimsup3  42449  climliminflimsup4  42450  xlimliminflimsup  42501  preimaioomnf  43351  incsmf  43373  issmfle  43376  decsmf  43397
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