Theorem fssdmd 6728

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 6720	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
4	1, 3	sseqtrd 3974	1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3906  dom cdm 5663  ⟶wf 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-ss 3923  df-fn 6543  df-f 6544

This theorem is used by:  ordtypelem7  9489  vdwlem11  17068  gsumzoppg  20037  taylfvallem1  26549  taylply2  26560  taylply  26561  dvtaylp  26562  dvntaylp0  26564  taylthlem1  26565  taylthlem2  26566  tocyccntz  33487  omssubadd  34714