Theorem fssdmd 5209

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 5202	. 2 |- ( ph -> dom F = A )
4	1, 3	sseqtrd 3077	1 |- ( ph -> D C_ A )

Syntax hints:   -> wi 4   C_ wss 3013   dom cdm 4467   -->wf 5045

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-in 3019  df-ss 3026  df-fn 5052  df-f 5053

This theorem is referenced by: (None)