Theorem fssdmd 6617

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 6609	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
4	1, 3	sseqtrd 3966	1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)

Syntax hints:   → wi 4   ⊆ wss 3892  dom cdm 5590  ⟶wf 6428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-in 3899  df-ss 3909  df-fn 6435  df-f 6436

This theorem is referenced by:  ordtypelem7  9261  vdwlem11  16690  gsumzoppg  19543  taylfvallem1  25514  taylply2  25525  taylply  25526  dvtaylp  25527  dvntaylp0  25529  taylthlem1  25530  taylthlem2  25531  tocyccntz  31407  omssubadd  32263