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Theorem cosselcnvrefrels2 36214
 Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 25-Aug-2021.)
Assertion
Ref Expression
cosselcnvrefrels2 ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ))

Proof of Theorem cosselcnvrefrels2
StepHypRef Expression
1 elcnvrefrels2 36210 . 2 ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ ≀ 𝑅 ∈ Rels ))
2 cossssid 36147 . . 3 ( ≀ 𝑅 ⊆ I ↔ ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)))
32anbi1i 626 . 2 (( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ) ↔ ( ≀ 𝑅 ⊆ ( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ∧ ≀ 𝑅 ∈ Rels ))
41, 3bitr4i 281 1 ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2111   ∩ cin 3857   ⊆ wss 3858   I cid 5429   × cxp 5522  dom cdm 5524  ran crn 5525   ≀ ccoss 35893   Rels crels 35895   CnvRefRels ccnvrefrels 35901 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 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-coss 36099  df-rels 36165  df-ssr 36178  df-cnvrefs 36203  df-cnvrefrels 36204 This theorem is referenced by:  cosselcnvrefrels3  36215  cosselcnvrefrels4  36216  cosselcnvrefrels5  36217  dffunsALTV2  36357  elfunsALTV2  36366  dfdisjs2  36382  eldisjs2  36396
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