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Theorem cnvrefrelcoss2 36084
 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 35979 . . 3 Rel ≀ 𝑅
2 dfcnvrefrel2 36079 . . 3 ( CnvRefRel ≀ 𝑅 ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ Rel ≀ 𝑅))
31, 2mpbiran2 709 . 2 ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
4 cossssid 36018 . 2 ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
53, 4bitr4i 281 1 ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∩ cin 3882   ⊆ wss 3883   I cid 5428   × cxp 5521  dom cdm 5523  ran crn 5524  Rel wrel 5528   ≀ ccoss 35764   CnvRefRel wcnvrefrel 35773 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-coss 35970  df-cnvrefrel 36076 This theorem is referenced by:  dffunALTV2  36232  funALTVfun  36242  dfdisjALTV2  36258
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