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Theorem refrelid 37030
Description: Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.)
Assertion
Ref Expression
refrelid RefRel I

Proof of Theorem refrelid
StepHypRef Expression
1 ssid 3967 . 2 ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I ))
2 reli 5783 . 2 Rel I
3 df-refrel 37020 . 2 ( RefRel I ↔ (( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) ∧ Rel I ))
41, 2, 3mpbir2an 710 1 RefRel I
Colors of variables: wff setvar class
Syntax hints:  cin 3910  wss 3911   I cid 5531   × cxp 5632  dom cdm 5634  ran crn 5635  Rel wrel 5639   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-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-ss 3928  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-refrel 37020
This theorem is referenced by: (None)
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