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Theorem refrelcoss 35306
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 35248 . 2 (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅)
2 dfrefrel2 35299 . 2 ( RefRel ≀ 𝑅 ↔ (( I ∩ (dom ≀ 𝑅 × ran ≀ 𝑅)) ⊆ ≀ 𝑅 ∧ Rel ≀ 𝑅))
31, 2mpbir 232 1 RefRel ≀ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wa 396  cin 3860  wss 3861   I cid 5350   × cxp 5444  dom cdm 5446  ran crn 5447  Rel wrel 5451  ccoss 34998   RefRel wrefrel 35004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pr 5224
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-br 4965  df-opab 5027  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-coss 35203  df-refrel 35296
This theorem is referenced by:  eqvrelcoss  35396
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