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Theorem relcoels 34608
Description: Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.)
Assertion
Ref Expression
relcoels Rel ∼ 𝐴

Proof of Theorem relcoels
StepHypRef Expression
1 relcoss 34607 . 2 Rel ≀ ( E ↾ 𝐴)
2 df-coels 34599 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32releqi 5372 . 2 (Rel ∼ 𝐴 ↔ Rel ≀ ( E ↾ 𝐴))
41, 3mpbir 222 1 Rel ∼ 𝐴
Colors of variables: wff setvar class
Syntax hints:   E cep 5189  ccnv 5276  cres 5279  Rel wrel 5282  ccoss 34404  ccoels 34405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-opab 4872  df-xp 5283  df-rel 5284  df-coss 34598  df-coels 34599
This theorem is referenced by: (None)
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