Theorem frexr 46340

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 11334	. . 3 ⊢ ℝ ⊆ ℝ*
3	2	a1i 11	. 2 ⊢ (𝜑 → ℝ ⊆ ℝ*)
4	1, 3	fssd 6719	1 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3899  ⟶wf 6527  ℝcr 11180  ℝ^*cxr 11323

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 2733

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 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-ss 3916  df-f 6535  df-xr 11328

This theorem is used by:  limsupubuz  46667  limsupreuz  46691  limsupvaluz2  46692  supcnvlimsup  46694  limsupgtlem  46731  liminflimsupclim  46761  climliminflimsup2  46763  climliminflimsup3  46764  climliminflimsup4  46765  xlimliminflimsup  46816  hoicvr  47502  preimaioomnf  47673  incsmf  47696  issmfle  47699  decsmf  47721  smfsupdmmbllem  47798  smfinfdmmbllem  47802