Theorem frexr 45814

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 11189	. . 3 ⊢ ℝ ⊆ ℝ*
3	2	a1i 11	. 2 ⊢ (𝜑 → ℝ ⊆ ℝ*)
4	1, 3	fssd 6685	1 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

Syntax hints:   → wi 4   ⊆ wss 3889  ⟶wf 6494  ℝcr 11037  ℝ^*cxr 11178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-un 3894  df-ss 3906  df-f 6502  df-xr 11183

This theorem is referenced by:  limsupubuz  46141  limsupreuz  46165  limsupvaluz2  46166  supcnvlimsup  46168  limsupgtlem  46205  liminflimsupclim  46235  climliminflimsup2  46237  climliminflimsup3  46238  climliminflimsup4  46239  xlimliminflimsup  46290  hoicvr  46976  preimaioomnf  47147  incsmf  47170  issmfle  47173  decsmf  47195  smfsupdmmbllem  47272  smfinfdmmbllem  47276