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Theorem fssdmd 6727
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 6719 . 2 (𝜑 → dom 𝐹 = 𝐴)
41, 3sseqtrd 4015 1 (𝜑𝐷𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3941  dom cdm 5667  wf 6530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958  df-fn 6537  df-f 6538
This theorem is referenced by:  ordtypelem7  9516  vdwlem11  16929  gsumzoppg  19860  taylfvallem1  26234  taylply2  26245  taylply  26246  dvtaylp  26247  dvntaylp0  26249  taylthlem1  26250  taylthlem2  26251  tocyccntz  32797  omssubadd  33819  gg-taylthlem2  35684
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