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Theorem refrelcoss 38051
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 37992 . 2 (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
2 dfrefrel2 38043 . 2 ( RefRel ≀ 𝑅 ↔ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅))
31, 2mpbir 230 1 RefRel ≀ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 394  cin 3938  wss 3939   I cid 5569   × cxp 5670  dom cdm 5672  ran crn 5673  Rel wrel 5677  ccoss 37705   RefRel wrefrel 37711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-coss 37939  df-refrel 38040
This theorem is referenced by:  eqvrelcoss  38145
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