Theorem dfcnvrefrel4 39116

Description: Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024.)

Assertion

Ref	Expression
dfcnvrefrel4	⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))

Proof of Theorem dfcnvrefrel4

Step	Hyp	Ref	Expression
1		df-cnvrefrel 39111	. 2 ⊢ ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2		cnvref4 38854	. 2 ⊢ (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅 ⊆ I ))
3	1, 2	bianim 584	1 ⊢ ( CnvRefRel 𝑅 ↔ (𝑅 ⊆ I ∧ Rel 𝑅))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∩ cin 3904   ⊆ wss 3905   I cid 5542   × cxp 5646  dom cdm 5648  ran crn 5649  Rel wrel 5653   CnvRefRel wcnvrefrel 38696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-rel 5655  df-cnv 5656  df-dm 5658  df-rn 5659  df-res 5660  df-cnvrefrel 39111

This theorem is referenced by:  dfcnvrefrel5  39117  dfantisymrel4  39368