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Theorem ffdmd 6212
 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 6211 . . 3 (𝐹:𝐴𝐵 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
31, 2syl 17 . 2 (𝜑 → (𝐹:dom 𝐹𝐵 ∧ dom 𝐹𝐴))
43simpld 477 1 (𝜑𝐹:dom 𝐹𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ⊆ wss 3703  dom cdm 5254  ⟶wf 6033 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-in 3710  df-ss 3717  df-fn 6040  df-f 6041 This theorem is referenced by:  upgrres1  26375  umgr2v2e  26602  pliguhgr  27620  xlimmnfvlem1  40530  xlimpnfvlem1  40534  issmfd  41419  issmfdf  41421  cnfsmf  41424  issmfled  41441  issmfgtd  41444  smfsuplem1  41492
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