Theorem dfrefrel5 39060

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 39058	. 2 ⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ∧ Rel 𝑅))
2		ref5 38782	. 2 ⊢ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ 𝑅 ↔ ∀𝑥 ∈ (dom 𝑅 ∩ ran 𝑅)𝑥𝑅𝑥)
3	1, 2	bianbi 636	1 ⊢ ( RefRel 𝑅 ↔ (∀𝑥 ∈ (dom 𝑅 ∩ ran 𝑅)𝑥𝑅𝑥 ∧ Rel 𝑅))

Syntax hints:   ↔ wb 208   ∧ wa 399  ∀wral 3075   ∩ cin 3903   ⊆ wss 3904   class class class wbr 5099   I cid 5539   × cxp 5643  dom cdm 5645  ran crn 5646  Rel wrel 5650   RefRel wrefrel 38652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-refrel 39055

This theorem is referenced by:  refrelressn  39067