Theorem refrelressn 38971

Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38899) is reflexive. (Contributed by Peter Mazsa, 12-Jun-2024.)

Assertion

Ref	Expression
refrelressn	⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴}))

Proof of Theorem refrelressn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		refressn 38900	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
2		relres 5957	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dfrefrel5 38964	. 2 ⊢ ( RefRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥 ∧ Rel (𝑅 ↾ {𝐴})))
4	1, 2, 3	sylanblrc 596	1 ⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴}))

Syntax hints:   → wi 4   ∈ wcel 2119  ∀wral 3053   ∩ cin 3882  {csn 4555   class class class wbr 5072  dom cdm 5618  ran crn 5619   ↾ cres 5620  Rel wrel 5623   RefRel wrefrel 38556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-refrel 38959

This theorem is referenced by: (None)