Theorem dfrefrel5 38553

Description: Alternate definition of the reflexive relation predicate. (Contributed by Peter Mazsa, 12-Dec-2023.)

Assertion

Ref	Expression
dfrefrel5	⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ (dom 𝑅 ∩ ran 𝑅)𝑥𝑅𝑥 ∧ Rel 𝑅))

Distinct variable group:   𝑥,𝑅

Proof of Theorem dfrefrel5

Step	Hyp	Ref	Expression
1		dfrefrel2 38551	. 2 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2		ref5 38346	. 2 ⊢ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ∀𝑥 ∈ (dom 𝑅 ∩ ran 𝑅)𝑥𝑅𝑥)
3	1, 2	bianbi 627	1 ⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ (dom 𝑅 ∩ ran 𝑅)𝑥𝑅𝑥 ∧ Rel 𝑅))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wral 3047   ∩ cin 3901   ⊆ wss 3902   class class class wbr 5091   I cid 5510   × cxp 5614  dom cdm 5616  ran crn 5617  Rel wrel 5621   RefRel wrefrel 38220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-refrel 38548

This theorem is referenced by:  refrelressn  38560