Theorem fssdm 5460

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 5452	. 2 |- ( ph -> dom F = A )
4	1, 3	sseqtrid 3251	1 |- ( ph -> D C_ A )

Syntax hints:   -> wi 4   C_ wss 3174   dom cdm 4693   -->wf 5286

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 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187  df-fn 5293  df-f 5294

This theorem is referenced by:  fisumss  11818  fprodssdc  12016  ghmpreima  13717  cnclima  14810  txcnmpt  14860  xmeter  15023