Theorem refrelressn 37850

Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 37768) 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 37769	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
2		relres 6000	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dfrefrel5 37843	. 2 ⊢ ( RefRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥 ∧ Rel (𝑅 ↾ {𝐴})))
4	1, 2, 3	sylanblrc 589	1 ⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴}))

Syntax hints:   → wi 4   ∈ wcel 2098  ∀wral 3053   ∩ cin 3939  {csn 4620   class class class wbr 5138  dom cdm 5666  ran crn 5667   ↾ cres 5668  Rel wrel 5671   RefRel wrefrel 37505

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-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

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-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-refrel 37838

This theorem is referenced by: (None)