Theorem frexr 45836

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	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)

Assertion

Ref	Expression
frexr	⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

Proof of Theorem frexr

Step	Hyp	Ref	Expression
1		frexr.1	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
2		ressxr 11187	. . 3 ⊢ ℝ ⊆ ℝ*
3	2	a1i 11	. 2 ⊢ (𝜑 → ℝ ⊆ ℝ*)
4	1, 3	fssd 6679	1 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

Syntax hints:   → wi 4   ⊆ wss 3890  ⟶wf 6488  ℝcr 11035  ℝ^*cxr 11176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-ss 3907  df-f 6496  df-xr 11181

This theorem is referenced by:  limsupubuz  46163  limsupreuz  46187  limsupvaluz2  46188  supcnvlimsup  46190  limsupgtlem  46227  liminflimsupclim  46257  climliminflimsup2  46259  climliminflimsup3  46260  climliminflimsup4  46261  xlimliminflimsup  46312  hoicvr  46998  preimaioomnf  47169  incsmf  47192  issmfle  47195  decsmf  47217  smfsupdmmbllem  47294  smfinfdmmbllem  47298