Theorem fssdm 5487

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 ⊆ dom 𝐹
fssdm.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
fssdm	⊢ (𝜑 → 𝐷 ⊆ 𝐴)

Proof of Theorem fssdm

Step	Hyp	Ref	Expression
1		fssdm.d	. 2 ⊢ 𝐷 ⊆ dom 𝐹
2		fssdm.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 5479	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
4	1, 3	sseqtrid 3274	1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)

Syntax hints:   → wi 4   ⊆ wss 3197  dom cdm 4718  ⟶wf 5313

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-fn 5320  df-f 5321

This theorem is referenced by:  fisumss  11898  fprodssdc  12096  ghmpreima  13798  cnclima  14891  txcnmpt  14941  xmeter  15104