Theorem fssdm 5544

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

Hypotheses

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

Assertion

Ref	Expression
fssdm	⊢ (𝜑 → 𝐷 ⊆ 𝐴)

Proof of Theorem fssdm

Step	Hyp	Ref	Expression
1		fssdm.d	. 2 ⊢ 𝐷 ⊆ dom 𝐹
2		fssdm.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	fdmd 5535	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
4	1, 3	sseqtrid 3298	1 ⊢ (𝜑 → 𝐷 ⊆ 𝐴)

Syntax hints:   → wi 4   ⊆ wss 3220  dom cdm 4769  ⟶wf 5368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-fn 5375  df-f 5376

This theorem is referenced by:  fisumss  12137  fprodssdc  12335  ghmpreima  14046  cnclima  15247  txcnmpt  15297  xmeter  15460