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Theorem cnvrefrelcoss2 38538
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 38424 . . 3 Rel ≀ 𝑅
2 dfcnvrefrel2 38531 . . 3 ( CnvRefRel ≀ 𝑅 ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ Rel ≀ 𝑅))
31, 2mpbiran2 710 . 2 ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
4 cossssid 38468 . 2 ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
53, 4bitr4i 278 1 ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I )
Colors of variables: wff setvar class
Syntax hints:  wb 206  cin 3950  wss 3951   I cid 5577   × cxp 5683  dom cdm 5685  ran crn 5686  Rel wrel 5690  ccoss 38182   CnvRefRel wcnvrefrel 38191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-coss 38412  df-cnvrefrel 38528
This theorem is referenced by:  dffunALTV2  38689  funALTVfun  38699  dfdisjALTV2  38715
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