Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  refrelid Structured version   Visualization version   GIF version

Theorem refrelid 35866
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 3975 . 2 ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I ))
2 reli 5685 . 2 Rel I
3 df-refrel 35857 . 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 3918  wss 3919   I cid 5446   × cxp 5540  dom cdm 5542  ran crn 5543  Rel wrel 5547   RefRel wrefrel 35564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-refrel 35857
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator