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Theorem fssdmd 5375
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 5368 . 2 (𝜑 → dom 𝐹 = 𝐴)
41, 3sseqtrd 3193 1 (𝜑𝐷𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3129  dom cdm 4623  wf 5208
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142  df-fn 5215  df-f 5216
This theorem is referenced by: (None)
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