Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frexr Structured version   Visualization version   GIF version

Theorem frexr 39086
 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 10030 . . 3 ℝ ⊆ ℝ*
32a1i 11 . 2 (𝜑 → ℝ ⊆ ℝ*)
41, 3fssd 6016 1 (𝜑𝐹:𝐴⟶ℝ*)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3556  ⟶wf 5845  ℝcr 9882  ℝ*cxr 10020 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3561  df-in 3563  df-ss 3570  df-f 5853  df-xr 10025 This theorem is referenced by:  limsupubuz  39367  limsupreuz  39391  limsupvaluz2  39392  supcnvlimsup  39394  preimaioomnf  40252  incsmf  40274  issmfle  40277  decsmf  40298
 Copyright terms: Public domain W3C validator