Theorem refrelressn 39199

Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 39127) 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 39128	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
2		relres 6004	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dfrefrel5 39192	. 2 ⊢ ( RefRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥 ∧ Rel (𝑅 ↾ {𝐴})))
4	1, 2, 3	sylanblrc 601	1 ⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴}))

Syntax hints:   → wi 4   ∈ wcel 2141  ∀wral 3077   ∩ cin 3903  {csn 4588   class class class wbr 5108  dom cdm 5661  ran crn 5662   ↾ cres 5663  Rel wrel 5666   RefRel wrefrel 38784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-refrel 39187

This theorem is referenced by: (None)