Theorem refrelressn 38522

Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 38440) 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 38441	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
2		relres 5979	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dfrefrel5 38515	. 2 ⊢ ( RefRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥 ∧ Rel (𝑅 ↾ {𝐴})))
4	1, 2, 3	sylanblrc 590	1 ⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴}))

Syntax hints:   → wi 4   ∈ wcel 2109  ∀wral 3045   ∩ cin 3916  {csn 4592   class class class wbr 5110  dom cdm 5641  ran crn 5642   ↾ cres 5643  Rel wrel 5646   RefRel wrefrel 38182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-refrel 38510

This theorem is referenced by: (None)