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Theorem eldmressn 46987
Description: Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Assertion
Ref Expression
eldmressn (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)

Proof of Theorem eldmressn
StepHypRef Expression
1 elin 3979 . . 3 (𝐵 ∈ ({𝐴} ∩ dom 𝐹) ↔ (𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹))
2 elsni 4648 . . . 4 (𝐵 ∈ {𝐴} → 𝐵 = 𝐴)
32adantr 480 . . 3 ((𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹) → 𝐵 = 𝐴)
41, 3sylbi 217 . 2 (𝐵 ∈ ({𝐴} ∩ dom 𝐹) → 𝐵 = 𝐴)
5 dmres 6032 . 2 dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹)
64, 5eleq2s 2857 1 (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cin 3962  {csn 4631  dom cdm 5689  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-res 5701
This theorem is referenced by:  dfdfat2  47078
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