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Theorem cnvrefrelcoss2 34598
 Description: Necessary and sufficient condition for a coset relation to be a converse reflexive relation. (Contributed by Peter Mazsa, 27-Jul-2021.)
Assertion
Ref Expression
cnvrefrelcoss2 ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I )

Proof of Theorem cnvrefrelcoss2
StepHypRef Expression
1 relcoss 34493 . . 3 Rel ≀ 𝑅
2 dfcnvrefrel2 34593 . . 3 ( CnvRefRel ≀ 𝑅 ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ Rel ≀ 𝑅))
31, 2mpbiran2 992 . 2 ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
4 cossssid 34532 . 2 ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
53, 4bitr4i 267 1 ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∩ cin 3706   ⊆ wss 3707   I cid 5165   × cxp 5256  dom cdm 5258  ran crn 5259  Rel wrel 5263   ≀ ccoss 34288   CnvRefRel wcnvrefrel 34297 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-iun 4666  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-fun 6043  df-fn 6044  df-coss 34484  df-cnvrefrel 34590 This theorem is referenced by: (None)
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