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Theorem cosselcnvrefrels3 36208
 Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 30-Aug-2021.)
Assertion
Ref Expression
cosselcnvrefrels3 ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑅 ∈ Rels ))
Distinct variable group:   𝑢,𝑅,𝑥,𝑦

Proof of Theorem cosselcnvrefrels3
StepHypRef Expression
1 cosselcnvrefrels2 36207 . 2 ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ))
2 cossssid3 36142 . . 3 ( ≀ 𝑅 ⊆ I ↔ ∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦))
32anbi1i 627 . 2 (( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ) ↔ (∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑅 ∈ Rels ))
41, 3bitri 278 1 ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑢𝑥𝑦((𝑢𝑅𝑥𝑢𝑅𝑦) → 𝑥 = 𝑦) ∧ ≀ 𝑅 ∈ Rels ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1537   = wceq 1539   ∈ wcel 2112   ⊆ wss 3859   class class class wbr 5033   I cid 5430   ≀ ccoss 35886   Rels crels 35888   CnvRefRels ccnvrefrels 35894 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-coss 36092  df-rels 36158  df-ssr 36171  df-cnvrefs 36196  df-cnvrefrels 36197 This theorem is referenced by:  dffunsALTV3  36351  elfunsALTV3  36360
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