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Theorem fssdm 5392
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 (𝜑𝐹:𝐴𝐵)
Assertion
Ref Expression
fssdm (𝜑𝐷𝐴)

Proof of Theorem fssdm
StepHypRef Expression
1 fssdm.d . 2 𝐷 ⊆ dom 𝐹
2 fssdm.f . . 3 (𝜑𝐹:𝐴𝐵)
32fdmd 5384 . 2 (𝜑 → dom 𝐹 = 𝐴)
41, 3sseqtrid 3217 1 (𝜑𝐷𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3141  dom cdm 4638  wf 5224
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-in 3147  df-ss 3154  df-fn 5231  df-f 5232
This theorem is referenced by:  fisumss  11414  fprodssdc  11612  ghmpreima  13160  cnclima  14076  txcnmpt  14126  xmeter  14289
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