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Theorem eldmressn 46319
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 3959 . . 3 (𝐵 ∈ ({𝐴} ∩ dom 𝐹) ↔ (𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹))
2 elsni 4640 . . . 4 (𝐵 ∈ {𝐴} → 𝐵 = 𝐴)
32adantr 480 . . 3 ((𝐵 ∈ {𝐴} ∧ 𝐵 ∈ dom 𝐹) → 𝐵 = 𝐴)
41, 3sylbi 216 . 2 (𝐵 ∈ ({𝐴} ∩ dom 𝐹) → 𝐵 = 𝐴)
5 dmres 5997 . 2 dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹)
64, 5eleq2s 2845 1 (𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cin 3942  {csn 4623  dom cdm 5669  cres 5671
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-dm 5679  df-res 5681
This theorem is referenced by:  dfdfat2  46408
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