Theorem fssdmd 5421

Description: Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, deduction form. (Contributed by AV, 21-Aug-2022.)

Hypotheses

Ref	Expression
fssdmd.f	|- ( ph -> F : A --> B )
fssdmd.d	|- ( ph -> D C_ dom F )

Assertion

Ref	Expression
fssdmd	|- ( ph -> D C_ A )

Proof of Theorem fssdmd

Step	Hyp	Ref	Expression
1		fssdmd.d	. 2 |- ( ph -> D C_ dom F )
2		fssdmd.f	. . 3 |- ( ph -> F : A --> B )
3	2	fdmd 5414	. 2 |- ( ph -> dom F = A )
4	1, 3	sseqtrd 3221	1 |- ( ph -> D C_ A )

Syntax hints:   -> wi 4   C_ wss 3157   dom cdm 4663   -->wf 5254

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-fn 5261  df-f 5262

This theorem is referenced by: (None)