Theorem ffdmd 5454

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 5453	. . 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 3168   dom cdm 4680   -->wf 5273

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3174  df-ss 3181  df-fn 5280  df-f 5281

This theorem is referenced by:  upgr1edc  15764