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Theorem refrelressn 38525
Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38443) is reflexive. (Contributed by Peter Mazsa, 12-Jun-2024.)
Assertion
Ref Expression
refrelressn (𝐴𝑉 → RefRel (𝑅 ↾ {𝐴}))

Proof of Theorem refrelressn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 refressn 38444 . 2 (𝐴𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
2 relres 6023 . 2 Rel (𝑅 ↾ {𝐴})
3 dfrefrel5 38518 . 2 ( RefRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥 ∧ Rel (𝑅 ↾ {𝐴})))
41, 2, 3sylanblrc 590 1 (𝐴𝑉 → RefRel (𝑅 ↾ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wral 3061  cin 3950  {csn 4626   class class class wbr 5143  dom cdm 5685  ran crn 5686  cres 5687  Rel wrel 5690   RefRel wrefrel 38188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-refrel 38513
This theorem is referenced by: (None)
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