Theorem fssdmd 6709

Description: Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, deduction form. (Contributed by AV, 21-Aug-2022.)

Hypotheses

Ref	Expression
fssdmd.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fssdmd.d	⊢ (𝜑 → 𝐷 ⊆ dom 𝐹)

Assertion

Ref	Expression
fssdmd	⊢ (𝜑 → 𝐷 ⊆ 𝐴)

Proof of Theorem fssdmd

Step	Hyp	Ref	Expression
1		fssdmd.d	. 2 ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹)
2		fssdmd.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 6701	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
4	1, 3	sseqtrd 3986	1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)

Syntax hints:   → wi 4   ⊆ wss 3917  dom cdm 5641  ⟶wf 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2722  df-ss 3934  df-fn 6517  df-f 6518

This theorem is referenced by:  ordtypelem7  9484  vdwlem11  16969  gsumzoppg  19881  taylfvallem1  26271  taylply2  26282  taylply2OLD  26283  taylply  26284  dvtaylp  26285  dvntaylp0  26287  taylthlem1  26288  taylthlem2  26289  taylthlem2OLD  26290  tocyccntz  33108  omssubadd  34298