Theorem fssdmd 6754

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 6746	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
4	1, 3	sseqtrd 4035	1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)

Syntax hints:   → wi 4   ⊆ wss 3962  dom cdm 5688  ⟶wf 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-cleq 2726  df-ss 3979  df-fn 6565  df-f 6566

This theorem is referenced by:  ordtypelem7  9561  vdwlem11  17024  gsumzoppg  19976  taylfvallem1  26412  taylply2  26423  taylply2OLD  26424  taylply  26425  dvtaylp  26426  dvntaylp0  26428  taylthlem1  26429  taylthlem2  26430  taylthlem2OLD  26431  tocyccntz  33146  omssubadd  34281