Theorem fssdmd 6734

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 6726	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
4	1, 3	sseqtrd 4000	1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)

Syntax hints:   → wi 4   ⊆ wss 3931  dom cdm 5665  ⟶wf 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2726  df-ss 3948  df-fn 6544  df-f 6545

This theorem is referenced by:  ordtypelem7  9546  vdwlem11  17011  gsumzoppg  19930  taylfvallem1  26334  taylply2  26345  taylply2OLD  26346  taylply  26347  dvtaylp  26348  dvntaylp0  26350  taylthlem1  26351  taylthlem2  26352  taylthlem2OLD  26353  tocyccntz  33103  omssubadd  34261