Theorem ffdmd 5491

Description: The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
ffdmd.1	|- ( ph -> F : A --> B )

Assertion

Ref	Expression
ffdmd	|- ( ph -> F : dom F --> B )

Proof of Theorem ffdmd

Step	Hyp	Ref	Expression
1		ffdmd.1	. . 3 |- ( ph -> F : A --> B )
2		ffdm 5490	. . 3 |- ( F : A --> B -> ( F : dom F --> B /\ dom F C_ A ) )
3	1, 2	syl 14	. 2 |- ( ph -> ( F : dom F --> B /\ dom F C_ A ) )
4	3	simpld 112	1 |- ( ph -> F : dom F --> B )

Syntax hints:   -> wi 4   /\ wa 104   C_ wss 3197   dom cdm 4716   -->wf 5310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-fn 5317  df-f 5318

This theorem is referenced by:  upgr1edc  15906