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Theorem fssdmd 6519
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 (𝜑𝐹:𝐴𝐵)
fssdmd.d (𝜑𝐷 ⊆ dom 𝐹)
Assertion
Ref Expression
fssdmd (𝜑𝐷𝐴)

Proof of Theorem fssdmd
StepHypRef Expression
1 fssdmd.d . 2 (𝜑𝐷 ⊆ dom 𝐹)
2 fssdmd.f . . 3 (𝜑𝐹:𝐴𝐵)
32fdmd 6513 . 2 (𝜑 → dom 𝐹 = 𝐴)
41, 3sseqtrd 3993 1 (𝜑𝐷𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3919  dom cdm 5542  wf 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-in 3926  df-ss 3936  df-fn 6346  df-f 6347
This theorem is referenced by:  ordtypelem7  8985  vdwlem11  16325  gsumzoppg  19064  taylfvallem1  24955  taylply2  24966  taylply  24967  dvtaylp  24968  dvntaylp0  24970  taylthlem1  24971  taylthlem2  24972  tocyccntz  30818  omssubadd  31615
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