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Theorem fssdmd 6741
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 6733 . 2 (𝜑 → dom 𝐹 = 𝐴)
41, 3sseqtrd 4020 1 (𝜑𝐷𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3947  dom cdm 5678  wf 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-in 3954  df-ss 3964  df-fn 6551  df-f 6552
This theorem is referenced by:  ordtypelem7  9548  vdwlem11  16960  gsumzoppg  19899  taylfvallem1  26304  taylply2  26315  taylply2OLD  26316  taylply  26317  dvtaylp  26318  dvntaylp0  26320  taylthlem1  26321  taylthlem2  26322  taylthlem2OLD  26323  tocyccntz  32878  omssubadd  33920
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