Theorem frexr 46214

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 11277	. . 3 ⊢ ℝ ⊆ ℝ*
3	2	a1i 11	. 2 ⊢ (𝜑 → ℝ ⊆ ℝ*)
4	1, 3	fssd 6720	1 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3899  ⟶wf 6529  ℝcr 11123  ℝ^*cxr 11266

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904  df-ss 3916  df-f 6537  df-xr 11271

This theorem is used by:  limsupubuz  46541  limsupreuz  46565  limsupvaluz2  46566  supcnvlimsup  46568  limsupgtlem  46605  liminflimsupclim  46635  climliminflimsup2  46637  climliminflimsup3  46638  climliminflimsup4  46639  xlimliminflimsup  46690  hoicvr  47376  preimaioomnf  47547  incsmf  47570  issmfle  47573  decsmf  47595  smfsupdmmbllem  47672  smfinfdmmbllem  47676