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Theorem refrelcoss 37031
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 36972 . 2 (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
2 dfrefrel2 37023 . 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 397  cin 3910  wss 3911   I cid 5531   × cxp 5632  dom cdm 5634  ran crn 5635  Rel wrel 5639  ccoss 36680   RefRel wrefrel 36686
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-coss 36919  df-refrel 37020
This theorem is referenced by:  eqvrelcoss  37125
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