Theorem dfrefrel5 36731

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 36729	. 2 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2		ref5 36526	. 2 ⊢ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ∀𝑥 ∈ (dom 𝑅 ∩ ran 𝑅)𝑥𝑅𝑥)
3	1, 2	bianbi 36434	1 ⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ (dom 𝑅 ∩ ran 𝑅)𝑥𝑅𝑥 ∧ Rel 𝑅))

Syntax hints:   ↔ wb 205   ∧ wa 397  ∀wral 3061   ∩ cin 3891   ⊆ wss 3892   class class class wbr 5081   I cid 5499   × cxp 5598  dom cdm 5600  ran crn 5601  Rel wrel 5605   RefRel wrefrel 36387

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3333  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-dm 5610  df-rn 5611  df-res 5612  df-refrel 36726

This theorem is referenced by:  refrelressn  36738