Theorem ffdmd 5559

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 5558	. . 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   C_ wss 3220   dom cdm 4774   -->wf 5373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-fn 5380  df-f 5381

This theorem is used by:  upgr1edc  16374