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Theorem frexr 40121
Description: A function taking real values, is a function taking extended real values. Common case. (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 10296 . . 3 ℝ ⊆ ℝ*
32a1i 11 . 2 (𝜑 → ℝ ⊆ ℝ*)
41, 3fssd 6219 1 (𝜑𝐹:𝐴⟶ℝ*)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3716  wf 6046  cr 10148  *cxr 10286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-un 3721  df-in 3723  df-ss 3730  df-f 6054  df-xr 10291
This theorem is referenced by:  limsupubuz  40467  limsupreuz  40491  limsupvaluz2  40492  supcnvlimsup  40494  limsupgtlem  40531  liminfgelimsupuz  40542  liminflimsupclim  40561  preimaioomnf  41454  incsmf  41476  issmfle  41479  decsmf  41500
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