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Theorem refrelressn 37907
Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 37825) 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 37826 . 2 (𝐴𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
2 relres 6004 . 2 Rel (𝑅 ↾ {𝐴})
3 dfrefrel5 37900 . 2 ( RefRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥 ∧ Rel (𝑅 ↾ {𝐴})))
41, 2, 3sylanblrc 589 1 (𝐴𝑉 → RefRel (𝑅 ↾ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wral 3055  cin 3942  {csn 4623   class class class wbr 5141  dom cdm 5669  ran crn 5670  cres 5671  Rel wrel 5674   RefRel wrefrel 37562
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-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-refrel 37895
This theorem is referenced by: (None)
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