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Theorem cosselcnvrefrels5 35842
 Description: Necessary and sufficient condition for a coset relation to be an element of the converse reflexive relation class. (Contributed by Peter Mazsa, 5-Sep-2021.)
Assertion
Ref Expression
cosselcnvrefrels5 ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ∧ ≀ 𝑅 ∈ Rels ))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem cosselcnvrefrels5
StepHypRef Expression
1 cosselcnvrefrels2 35839 . 2 ( ≀ 𝑅 ∈ CnvRefRels ↔ ( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ))
2 cossssid5 35776 . . 3 ( ≀ 𝑅 ⊆ I ↔ ∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
32anbi1i 626 . 2 (( ≀ 𝑅 ⊆ I ∧ ≀ 𝑅 ∈ Rels ) ↔ (∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ∧ ≀ 𝑅 ∈ Rels ))
41, 3bitri 278 1 ( ≀ 𝑅 ∈ CnvRefRels ↔ (∀𝑥 ∈ ran 𝑅𝑦 ∈ ran 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ∧ ≀ 𝑅 ∈ Rels ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2115  ∀wral 3132   ∩ cin 3917   ⊆ wss 3918  ∅c0 4274   I cid 5442  ◡ccnv 5537  ran crn 5539  [cec 8272   ≀ ccoss 35518   Rels crels 35520   CnvRefRels ccnvrefrels 35526 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-ec 8276  df-coss 35724  df-rels 35790  df-ssr 35803  df-cnvrefs 35828  df-cnvrefrels 35829 This theorem is referenced by:  elfunsALTV5  35994
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