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Theorem ffdmd 6224
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 6223 . . 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 3715  dom cdm 5266  wf 6045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-in 3722  df-ss 3729  df-fn 6052  df-f 6053
This theorem is referenced by:  upgrres1  26404  umgr2v2e  26631  pliguhgr  27649  xlimmnfvlem1  40561  xlimpnfvlem1  40565  issmfd  41450  issmfdf  41452  cnfsmf  41455  issmfled  41472  issmfgtd  41475  smfsuplem1  41523
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