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Theorem frexr 45396
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 11305 . . 3 ℝ ⊆ ℝ*
32a1i 11 . 2 (𝜑 → ℝ ⊆ ℝ*)
41, 3fssd 6753 1 (𝜑𝐹:𝐴⟶ℝ*)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3951  wf 6557  cr 11154  *cxr 11294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-f 6565  df-xr 11299
This theorem is referenced by:  limsupubuz  45728  limsupreuz  45752  limsupvaluz2  45753  supcnvlimsup  45755  limsupgtlem  45792  liminflimsupclim  45822  climliminflimsup2  45824  climliminflimsup3  45825  climliminflimsup4  45826  xlimliminflimsup  45877  preimaioomnf  46734  incsmf  46757  issmfle  46760  decsmf  46782  smfsupdmmbllem  46859  smfinfdmmbllem  46863
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