Theorem fssdm 5352

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

Hypotheses

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

Assertion

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

Proof of Theorem fssdm

Step	Hyp	Ref	Expression
1		fssdm.d	. 2 |- D C_ dom F
2		fssdm.f	. . 3 |- ( ph -> F : A --> B )
3	2	fdmd 5344	. 2 |- ( ph -> dom F = A )
4	1, 3	sseqtrid 3192	1 |- ( ph -> D C_ A )

Syntax hints:   -> wi 4   C_ wss 3116   dom cdm 4604   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129  df-fn 5191  df-f 5192

This theorem is referenced by:  fisumss  11333  fprodssdc  11531  cnclima  12863  txcnmpt  12913  xmeter  13076