Theorem frexr 43167

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 11089	. . 3 ⊢ ℝ ⊆ ℝ*
3	2	a1i 11	. 2 ⊢ (𝜑 → ℝ ⊆ ℝ*)
4	1, 3	fssd 6653	1 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

Syntax hints:   → wi 4   ⊆ wss 3896  ⟶wf 6459  ℝcr 10940  ℝ^*cxr 11078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3443  df-un 3901  df-in 3903  df-ss 3913  df-f 6467  df-xr 11083

This theorem is referenced by:  limsupubuz  43498  limsupreuz  43522  limsupvaluz2  43523  supcnvlimsup  43525  limsupgtlem  43562  liminflimsupclim  43592  climliminflimsup2  43594  climliminflimsup3  43595  climliminflimsup4  43596  xlimliminflimsup  43647  preimaioomnf  44502  incsmf  44525  issmfle  44528  decsmf  44550