Theorem frexr 46489

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 11356	. . 3 ⊢ ℝ ⊆ ℝ*
3	2	a1i 11	. 2 ⊢ (𝜑 → ℝ ⊆ ℝ*)
4	1, 3	fssd 6727	1 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3895  ⟶wf 6534  ℝcr 11202  ℝ^*cxr 11345

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-un 4151  df-f 6542  df-xr 11350

This theorem is used by:  limsupubuz  46816  limsupreuz  46840  limsupvaluz2  46841  supcnvlimsup  46843  limsupgtlem  46880  liminflimsupclim  46910  climliminflimsup2  46912  climliminflimsup3  46913  climliminflimsup4  46914  xlimliminflimsup  46965  hoicvr  47651  preimaioomnf  47822  incsmf  47845  issmfle  47848  decsmf  47870  smfsupdmmbllem  47947  smfinfdmmbllem  47951