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Theorem refrelressn 38505
Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38423) 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 38424 . 2 (𝐴𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
2 relres 6025 . 2 Rel (𝑅 ↾ {𝐴})
3 dfrefrel5 38498 . 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 2105  wral 3058  cin 3961  {csn 4630   class class class wbr 5147  dom cdm 5688  ran crn 5689  cres 5690  Rel wrel 5693   RefRel wrefrel 38167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-refrel 38493
This theorem is referenced by: (None)
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