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Theorem refrelid 38026
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 4004 . 2 ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I ))
2 reli 5832 . 2 Rel I
3 df-refrel 38016 . 2 ( RefRel I ↔ (( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) ∧ Rel I ))
41, 2, 3mpbir2an 709 1 RefRel I
Colors of variables: wff setvar class
Syntax hints:  cin 3948  wss 3949   I cid 5579   × cxp 5680  dom cdm 5682  ran crn 5683  Rel wrel 5687   RefRel wrefrel 37687
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-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-refrel 38016
This theorem is referenced by: (None)
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