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Theorem fssdmd 6688
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 6680 . 2 (𝜑 → dom 𝐹 = 𝐴)
41, 3sseqtrd 3985 1 (𝜑𝐷𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3911  dom cdm 5634  wf 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-ss 3928  df-fn 6500  df-f 6501
This theorem is referenced by:  ordtypelem7  9465  vdwlem11  16868  gsumzoppg  19726  taylfvallem1  25732  taylply2  25743  taylply  25744  dvtaylp  25745  dvntaylp0  25747  taylthlem1  25748  taylthlem2  25749  tocyccntz  32042  omssubadd  32957
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