Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ffdmd Structured version   Visualization version   GIF version

Theorem ffdmd 5976
 Description: The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
ffdmd.1 (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
ffdmd (𝜑𝐹:dom 𝐹𝐵)

Proof of Theorem ffdmd
StepHypRef Expression
1 ffdmd.1 . . 3 (𝜑𝐹:𝐴𝐵)
2 ffdm 5975 . . 3 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
31, 2syl 17 . 2 (𝜑 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
43simpld 474 1 (𝜑𝐹:dom 𝐹𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ⊆ wss 3540  dom cdm 5038  ⟶wf 5800 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-in 3547  df-ss 3554  df-fn 5807  df-f 5808 This theorem is referenced by:  issmfd  39621  issmfdf  39624  cnfsmf  39627  issmfled  39644  issmfgtd  39647  umgr2v2e  40741
 Copyright terms: Public domain W3C validator