Theorem fssdm 5362

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 5354	. 2 |- ( ph -> dom F = A )
4	1, 3	sseqtrid 3197	1 |- ( ph -> D C_ A )

Syntax hints:   -> wi 4   C_ wss 3121   dom cdm 4611   -->wf 5194

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-fn 5201  df-f 5202

This theorem is referenced by:  fisumss  11355  fprodssdc  11553  cnclima  13017  txcnmpt  13067  xmeter  13230