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Theorem refrelcoss 35938
 Description: The class of cosets by 𝑅 is reflexive. (Contributed by Peter Mazsa, 4-Jul-2020.)
Assertion
Ref Expression
refrelcoss RefRel ≀ 𝑅

Proof of Theorem refrelcoss
StepHypRef Expression
1 refrelcoss2 35880 . 2 (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
2 dfrefrel2 35931 . 2 ( RefRel ≀ 𝑅 ↔ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅))
31, 2mpbir 234 1 RefRel ≀ 𝑅
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∩ cin 3880   ⊆ wss 3881   I cid 5424   × cxp 5517  dom cdm 5519  ran crn 5520  Rel wrel 5524   ≀ ccoss 35629   RefRel wrefrel 35635 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 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-coss 35835  df-refrel 35928 This theorem is referenced by:  eqvrelcoss  36028
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