Theorem dfrefrel5 39497

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 39495	. 2 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2		ref5 39219	. 2 ⊢ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ∀𝑥 ∈ (dom 𝑅 ∩ ran 𝑅)𝑥𝑅𝑥)
3	1, 2	bianbi 639	1 ⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ (dom 𝑅 ∩ ran 𝑅)𝑥𝑅𝑥 ∧ Rel 𝑅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∀wral 3077   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5103   I cid 5545   × cxp 5649  dom cdm 5651  ran crn 5652  Rel wrel 5656   RefRel wrefrel 39089

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-refrel 39492

This theorem is used by:  refrelressn  39504