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Theorem frexr 38545
 Description: A function taking real values, is a function taking extended real values. Common case. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
frexr.1 (𝜑𝐹:𝐴⟶ℝ)
Assertion
Ref Expression
frexr (𝜑𝐹:𝐴⟶ℝ*)

Proof of Theorem frexr
StepHypRef Expression
1 frexr.1 . 2 (𝜑𝐹:𝐴⟶ℝ)
2 ressxr 9962 . . 3 ℝ ⊆ ℝ*
32a1i 11 . 2 (𝜑 → ℝ ⊆ ℝ*)
41, 3fssd 5970 1 (𝜑𝐹:𝐴⟶ℝ*)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ⊆ wss 3540  ⟶wf 5800  ℝcr 9814  ℝ*cxr 9952 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-f 5808  df-xr 9957 This theorem is referenced by:  preimaioomnf  39606  issmfltle  39622  incsmf  39629  issmfle  39632  decsmf  39653
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