Theorem refrelressn 39739

Description: Any class ' R ' restricted to the singleton of the set ' A ' (see ressn2 39667) 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 39668	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥)
2		relres 6008	. 2 ⊢ Rel (𝑅 ↾ {𝐴})
3		dfrefrel5 39732	. 2 ⊢ ( RefRel (𝑅 ↾ {𝐴}) ↔ (∀𝑥 ∈ (dom (𝑅 ↾ {𝐴}) ∩ ran (𝑅 ↾ {𝐴}))𝑥(𝑅 ↾ {𝐴})𝑥 ∧ Rel (𝑅 ↾ {𝐴})))
4	1, 2, 3	sylanblrc 602	1 ⊢ (𝐴 ∈ 𝑉 → RefRel (𝑅 ↾ {𝐴}))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ∀wral 3080   ∩ cin 4092  {csn 4588   class class class wbr 5106  dom cdm 5656  ran crn 5657   ↾ cres 5658  Rel wrel 5661   RefRel wrefrel 39324

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-dm 5666  df-rn 5667  df-res 5668  df-refrel 39727

This theorem is used by: (None)