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Theorem cnvrefrelcoss2 35325
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 35220 . . 3 Rel ≀ 𝑅
2 dfcnvrefrel2 35320 . . 3 ( CnvRefRel ≀ 𝑅 ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ Rel ≀ 𝑅))
31, 2mpbiran2 706 . 2 ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
4 cossssid 35259 . 2 ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
53, 4bitr4i 279 1 ( CnvRefRel ≀ 𝑅 ↔ ≀ 𝑅 ⊆ I )
Colors of variables: wff setvar class
Syntax hints:  wb 207  cin 3864  wss 3865   I cid 5354   × cxp 5448  dom cdm 5450  ran crn 5451  Rel wrel 5455  ccoss 35006   CnvRefRel wcnvrefrel 35015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-coss 35211  df-cnvrefrel 35317
This theorem is referenced by:  dffunALTV2  35473  funALTVfun  35483  dfdisjALTV2  35499
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