Theorem fssdm 5440

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 5432	. 2 |- ( ph -> dom F = A )
4	1, 3	sseqtrid 3243	1 |- ( ph -> D C_ A )

Syntax hints:   -> wi 4   C_ wss 3166   dom cdm 4675   -->wf 5267

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-in 3172  df-ss 3179  df-fn 5274  df-f 5275

This theorem is referenced by:  fisumss  11703  fprodssdc  11901  ghmpreima  13602  cnclima  14695  txcnmpt  14745  xmeter  14908