Theorem fssdm 5245

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 5237	. 2 |- ( ph -> dom F = A )
4	1, 3	sseqtrid 3113	1 |- ( ph -> D C_ A )

Syntax hints:   -> wi 4   C_ wss 3037   dom cdm 4499   -->wf 5077

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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3043  df-ss 3050  df-fn 5084  df-f 5085

This theorem is referenced by:  fisumss  11053  cnclima  12234  txcnmpt  12284  xmeter  12425